WP 19/24

Reviewing the Existing Evidence of the Conditional Cash

Transfer in India through the Partial Identification Approach

Toshiaki Aizawa

October 2019

http://www.york.ac.uk/economics/postgrad/herc/hedg/wps/

HEDG HEALTH, ECONOMETRICS AND DATA GROUP

Reviewing the Existing Evidence of the Conditional Cash Transfer in

India through the Partial Identification Approach

Toshiaki Aizawa

Abstract

This paper re-estimates the causal impacts of a conditional cash transfer programme in India, the

Janani Suraksha Yojana (JSY), on maternal and child healthcare use. The main goal is to provide

new evidence and assess the validity of the identification assumptions employed in previous studies on

the JSY, through the conservative partial identification approach. We find that the average treatment

effects estimated under the conditional independence assumption lie outside the bound of the treatment

effect estimated under weaker but more credible assumptions, thereby suggesting that the selection bias

could not have been fully controlled for by the observable characteristics and that the average treatment

effects estimated in the previous studies may have been over- or under-estimated.

JEL codes: I12, I15, I18

Keywords: Conditional cash transfer; Partial identification; Conditional independence; India

1

1 Introduction

1.1 Background

The adequate use of healthcare services is an essential factor in successful maternal and child health

outcomes (Campbell and Graham, 2006; Chou et al., 2015; Darmstadt et al., 2009; Scott and Ronsmans,

2009). However, women in developing countries are often faced with multiple barriers to accessing health

services, and improvements in maternal healthcare access, especially by poor women, remain inadequate.

In India, only 51.2 per cent of women access antenatal care, and in 2014, only 38.7 per cent gave birth

in a health institution (UNICEF, 2018a). In addition, India has the highest neonatal mortality rate in

the world – as of 2015, around 20 per cent (=1.2 million) of global under-five deaths occurred in the

country (UNICEF, 2015, 2018b). Promoting institutional delivery and adequate use of antenatal care is

a key method to improve infant and neonatal mortalities (Lahariya, 2009; Pathak and Mohanty, 2010;

Freedman et al., 2007). However, women’s uptake of maternal care in India has been associated strongly

with wealth, and facilitating the adequate use of maternal care among poor and marginalised women is

still a major challenge (Pathak et al., 2010; Kesterton et al., 2010).

The success of conditional cash transfer (CCT) programmes in Latin America has led to an enthusiastically

mirrored response in Africa and Asia, and CCT has become one of the most adopted demand-side pro-

grammes to enhance healthcare use (Handa and Davis, 2006; Rawlings and Rubio, 2005). CCTs influence

health-seeking behaviours through financial incentives by transferring money to households contingent on

investments in human capital, such as health and education. Interest in using financial incentives to pro-

mote maternal and child healthcare utilisation has also spread to Southern Asia, which has low maternal

healthcare use and high neonatal mortality rates. CCT programmes intending to encourage maternity

and child healthcare use have been introduced in India, Nepal and Bangladesh (Devadasan et al., 2008).1

On 12 April 2005, the Indian government launched a nationwide CCT programme called Janani Suraksha

Yojana (JSY), or Safe Motherhood Programme, which promotes institutional delivery and receiving timely

antenatal and postnatal care to reduce infant and neonatal mortalities (Horton, 2010; Bhutta et al., 2010).

The JSY has become one of the largest cash transfer programmes in the world in terms of the number

of beneficiaries. The JSY provides a cash incentive to mothers that give birth in government-approved

1Pakistan and Cambodia also introduced a voucher-type programme with the same aims as the CCTs in South Asia(Jehan et al., 2012; Van de Poel et al., 2014).

2

health facilities2, and in this sense, it has a much narrower aim than most of the CCTs in Latin America,

which have multiple goals beyond maternal and child healthcare. Funded by central government, the JSY

is administered through the National Rural Health Mission. In 2014-2015, 10.4 million women benefited

from the scheme, representing a third of all women giving birth in the country annually (Ministry of

Health and Family Welfare, 2015).

Evidence in most previous studies has largely relied on inappropriate data or potentially untenable iden-

tification assumptions in order to yield a definitive causal impact. Until now, no study has ever tried

to explore the validity of the identification assumptions employed in previous studies on the JSY. The

main goal of this paper is to re-estimate the causal impact of the JSY participation on maternal and child

healthcare use, with weaker but more credible identification assumptions than used in previous studies.

Specifically, we non-parametrically identify the average treatment effect (ATE) by bounds. The partial

identification approach abandons point identification with strong assumptions and instead seeks causal

effects with much more credible assumptions (Manski, 1990, 1997; Manski and Pepper, 2000). In this

sense, the partial identification approach yields more conservative results.

In this paper, we perform inference under a spectrum of assumptions of varying identification power,

following the approach taken by Gundersen et al. (2012, 2017); Gonzalez (2005); Gerfin and Schellhorn

(2006) and Kreider et al. (2012). Any point-estimate obtained under the conventionally imposed strong

assumptions should lie within the bounds if these assumptions are valid. In this study, we show that

the causal impacts based on the assumptions employed in previous studies are larger/smaller than the

upper/lower limits of the ATE bounds, suggesting that the existing evidence could have been over-

estimated/under-estimated.

1.2 Janani Suraksha Yojana (JSY)

In the JSY scheme, women of 19 years of age and above are eligible for a cash benefit if they have a

below-the-poverty-line card issued by the government or belong to a scheduled caste or tribe. They can

receive 600 Indian rupees in urban areas and 700 rupees in rural areas after delivery in a public health

facility for their first two live births. The JSY does not cover the actual cost of institutional delivery and

maternal healthcare. The basic JSY scheme used to be the same across the country, but after November

2This includes public hospitals and accredited private institutions.

3

Figure 1: State types and JSY participation rate in 2010-16

Source: Author’s calculation from the National Family Health Survey.Note: LPS=Low performing state; and HPS=High performing state.

2006, different eligibility criteria and cash transfer amounts began to be applied in ten states with high

levels of maternal mortality and low levels of institutional delivery (i.e., Uttar Pradesh, Uttaranchal,

Bihar, Jharkhand, Madhya Pradesh, Chhattisgarh, Assam, Rajasthan, Odisha and Jammu and Kashmir)

(Bredenkamp, 2009). In these ten low-performing states (LPSs), the JSY provides a cash incentive to

all women regardless of age, numbers of children or socio-economic status. In other words, every woman

who gives birth in a public facility is eligible to receive a cash benefit. Moreover, in the LPSs, a higher

cash incentive is provided, namely 1,000 rupees in urban areas and 1,400 rupees in rural areas.3 In other

states that are classified as high-performing states (HPSs), the original eligibility criteria, as well as the

same cash incentives for pregnant mothers, continue to be applied. The left panel of Figure 1 illustrates

the HPSs and LPSs in the country and the right panel of Figure 1 shows the participation rates of the

JSY across the states.

The JSY addresses both the demand and the supply constraints of maternal healthcare services, i.e., it also

has a supply-side component, in which community-level health workers are given incentive payments for

encouraging pregnant women to give birth at health facilities. Community-level health workers, known as

‘accredited social health activists’, are the first and most important point of contact for pregnant women.

These accredited social health activists identify and register all pregnant women, assist them in developing

3This is around 8-12 paid days off from minimum wage manual labour (Joshi and Sivaram, 2014).

4

their birth plans and arrange the JSY for them. Accredited social health activists are given additional

cash incentives to encourage mothers to complete postnatal care. Cash incentives given to health workers

are intended to reduce absenteeism and enhance their overall performance.

2 Related literature

2.1 Evidence of the impacts of the JSY on maternal healthcare use

The positive impacts of CCTs in Latin America on maternal healthcare utilisation are reported by nu-

merous studies (Ranganathan and Lagarde, 2012; Barber and Gertler, 2010; Morris et al., 2004). For

systematic literature reviews of the CCT programme, see Glassman et al. (2013); Lagarde et al. (2007);

Bassett (2008) and Gaarder et al. (2010). In Mexico and El Salvador, for example, birth attendance by

skilled personnel increased by 11.4 percentage points and 12.3 percentage points respectively (Urquieta

et al., 2009; de Brauw and Peterman, 2011). However, different from CCTs which have much broader

aims beyond maternal healthcare, studies on those with narrower aims, such as the JSY, are relatively

sparse. Most of the existing literature explored the descriptive associations between JSY receipt and

healthcare use and health (Gupta et al., 2012, 2011; Thongkong et al., 2017; Randive et al., 2014, 2013;

Mukherjee and Singh, 2018; Purohit et al., 2014; Ng et al., 2014; Gopalan and Varatharajan, 2012), while

the studies estimating the causal impacts of the JSY are rather limited. This is mainly because the rigor-

ous randomised controlled trial (RCT) designed for the evaluation of the programme was not conducted

in India (Joshi and Sivaram, 2014).

Lim et al. (2010) is the first seminal study estimating the causal impact of JSY from the observational

data. Specifically, Lim et al. (2010) estimate the causal impacts on institutional delivery, use of antenatal

care with the individual-level and district-level data. According to their estimates with matching, the JSY

increases institutional deliveries by 43.5 percentage points and skilled birth attendance by 36.6 percentage

points. In addition, they report a significant increase in the use of antenatal care by 10.7 percentage points.

After the study by Lim et al. (2010), a number of studies extend analyses; they explore heterogeneity

in average treatment effect (ATE) across the population and estimate the impacts on the use of various

healthcare services (Carvalho et al., 2014). As well as the increase in maternal healthcare utilisation, re-

ductions in neonatal mortality and increases in the use of child healthcare are reported by Sengupta and

5

Sinha (2018), in which authors estimate the causal impacts with inverse probability weighting regression

approach. In addition, improvements in the mental health of beneficiary mothers (Powell-Jackson et al.,

2016), in breastfeeding and more pregnancies (Powell-Jackson et al., 2015; Nandi and Laxminarayan,

2016) are reported as additional – unintended – benefits.

Most studies exploit the second and the third waves of the District Level Household and Facility Survey

(DLHS-2 and DLHS-3), of which data were collected in 2002-2004 and 2007-2008. DLHS-2 is the survey

conducted before the JSY was launched, and DLHS-3 was conducted in the initial stage of the JSY imple-

mentation. Despite the introduction of JSY in 2005, its proper implementation actually started in 2007

(Das et al., 2011). Das et al. (2011) argues that in DLHS-3, many women who gave birth before 2007 were

misclassified as JSY beneficiaries and only a smaller proportion of women knew about the programme

itself at that time. Also, during the first few years of the implementation, many institutions were not

adequately prepared for maternal and child healthcare (Gopichandran and Chetlapalli, 2012). It turned

out that the DLHS-3 was not appropriate for causal analysis, and we had to wait for new data to become

available in order to estimate the causal effect of the JSY programme.

Recently, Rahman and Pallikadavath (2017) and Rahman and Pallikadavath (2018) have re-estimated

the impact of the JSY with the latest wave of the DLHS (DLHS-4), which was implemented in 2013-2014,

when JSY had matured enough to be known by almost all pregnant women. Rahman and Pallikada-

vath (2017) estimate the impacts of the JSY on various healthcare utilisations with the propensity score

matching (PSM). Rahman and Pallikadavath (2018) additionally estimate the impacts with the fuzzy

regression discontinuity design (RDD), which exploits the changes of eligibility for the JSY programme

with birth orders in HPSs. Major research into the causal effects of the JSY are listed in Table 1.

2.2 Identification assumptions in previous studies

Most of the existing literature which uses individual level data rests on strong assumptions in order to

point-identify the ATE. Previous studies on the JSY imposed the conditional independence assumption

(CIA), requiring that conditional on specific observable characteristics, the selection of the treatment

is random (Imbens and Wooldridge, 2009).4 Methods based on the CIA do not assume the possibility

that unobserved differences between participants and non-participants are associated with the difference

4A notable exception is Powell-Jackson et al. (2016), who collect their own data and exploit the fact that some womendid not receive the cash due to administrative problems in its disbursement.

6

Tab

le1:

Ma

jor

stu

die

ses

tim

atin

gth

eca

usa

leff

ects

ofth

eJS

YL

itera

ture

Data

Meth

ods/

Identi

ficati

on

sourc

eT

reatm

ent

Identi

fied

Main

outc

om

es

indic

ato

rE

ffect

Stu

dy

on

targ

ete

doutc

om

es

Lim

etal.

(2010)

DL

HS

2-3

Matc

hin

g,

Reg

ress

ion

JSY

part

icip

ati

on

AT

ED

eliv

ery,

bir

thatt

endance

,D

istr

ict-

level

diff

eren

ce-i

n-d

iffer

ence

sJSY

cover

age

rate

AT

Em

ort

ality

Carv

alh

oet

al.

(2014)

DL

HS

3P

rop

ensi

tysc

ore

matc

hin

gJSY

part

icip

ati

on

AT

EIm

munis

ati

on,

PN

C,

bre

ast

feed

ing

Sen

gupta

and

Sin

ha

(2018)

DL

HS

3O

LS

adju

sted

wit

hth

eIn

ver

seJSY

part

icip

ati

on

AT

EM

ort

ality

,st

illb

irth

,P

NC

,pro

babilit

yw

eighti

ng

imm

unis

ati

on

Rahm

an

and

Pallik

adav

ath

(2017)

DL

HS

4O

LS,

Pro

pen

sity

score

matc

hin

gJSY

part

icip

ati

on

AT

TD

eliv

ery,

AN

C,

PN

C,

imm

unis

ati

on

Rahm

an

and

Pallik

adav

ath

(2018)

DL

HS

4P

rop

ensi

tysc

ore

matc

hin

g,

JSY

part

icip

ati

on

AT

T,

Del

iver

y,A

NC

,P

NC

Fuzz

yre

gre

ssio

ndis

conti

nuit

ydes

ign

JSY

part

icip

ati

on

LA

TE

Josh

iand

Siv

ara

m(2

014)

DL

HS

2-3

Indiv

idual-

level

diff

eren

ce-i

n-d

iffer

ence

sJSY

elig

ibilit

yIT

TA

NC

,del

iver

y,P

NC

,bir

thatt

endance

Stu

dy

on

addit

ional

unin

tended

benefits

Pow

ell-

Jack

son

etal.

(2015)

DL

HS

2-3

Dis

tric

t-le

vel

diff

eren

ce-i

n-d

iffer

ence

sJSY

cover

age

rate

AT

EM

ort

ality

,del

iver

y,A

NC

,pre

gnancy

,bre

ast

feed

ing

Pow

ell-

Jack

son

etal.

(2016)

Ow

ndata

,Q

uasi

-exp

erim

enta

ldes

ign,

JSY

part

icip

ati

on

AT

EM

ate

rnal

men

tal

hea

lth

censu

sex

plo

itin

gth

eadm

inis

trati

ve

pro

ble

ms

inth

edis

burs

emen

t

Nandi

and

Laxm

inara

yan

(2016)

DL

HS

2-3

Pro

pen

sity

score

matc

hin

g+

JSY

part

icip

ati

on

AT

TF

erti

lity

diff

eren

ce-i

n-d

iffer

ence

s,ex

plo

itin

gth

ediff

eren

cein

the

ince

nti

ve

stru

cture

Note

:A

TE

=A

ver

age

trea

tmen

teff

ect;

AT

T=

Aver

age

trre

atm

ent

effec

ton

the

trea

ted;

ITT

=In

tenti

on-t

o-t

reat

effec

t;L

AT

E=

Loca

lav

erage

trea

tmen

teff

ect;

AN

C=

Ante

nata

lca

re;

and

PN

C=

Post

nata

lca

re.

7

in outcomes. However, participation in the JSY programme is likely to be dependent on individual un-

observable characteristics that could also affect healthcare use, such as the degree of being risk-averse.

Also, participants in the JSY may be more aware of the importance of maternal healthcare use and the

potential risk of delivery at home. If the receipt of the JSY is related to such unobservable characteristics,

the approaches based on the CIA still suffer selection bias and fail to estimate the causal impact.

Joshi and Sivaram (2014) estimated the intention-to-treat effect to deal with the selection problems that

had not been properly addressed, using the eligibility criteria, but we should note that the intention-to-

treat effect (ITT) is conceptually different from the treatment effects explored in other literature (Table

1). The district-level difference-in-differences (DID) is employed in Lim et al. (2010) and Powell-Jackson

et al. (2015), but the effect identified with the district-level DID is also conceptually different from the

ATEs identified in the other studies that rely on individual-level data (Table 1). As the district-level DID

exploits the variation in the roll-out of the JSY, it estimates the effect of percent point change in the JSY

coverage rate on the probability of healthcare use, not the effect of the JSY participation itself. Hence in

this study, we do not consider the ITT or the district-level DID.

The fuzzy RDD can deal with the individual unobservable characteristics, but the estimate is local in

the sense that it is the impact among those who are having a value of the running variable near the

cut-off value (Imbens and Angrist, 1994). Hence it is often difficult to extrapolate the impact for the en-

tire population from the estimated results among the sub-population. Rahman and Pallikadavath (2018)

use the birth order (parity) as a running variable and exploit the gap in the probability to participate

in the programme between mothers having one and two children and mothers having three to seven

children. However, the parity itself is highly likely to be endogenous. Hence, the observed discontinu-

ity in the probability of programme participation could be invalid as an exogenous source of identification.

In contrast to the existing literature, this study takes a totally different approach. Rather than relying on

strong yet questionable assumptions to achieve point-identification of the causal impact, we impose just

weak but credible identification assumptions to get the bound-estimation of the causal effect (Manski,

1990). The partial identification approach highlights what may be learned from the data without invoking

potentially untenable assumptions (Manski, 2003, 2013). By gradually adding assumptions, we explore

how much we can narrow the estimated bounds. The partial identification approach is very useful in the

8

situation in which a rigorous RCT was not implemented and hence valid instrumental variables to deal

with possible endogeneity are not available to researchers. The partial identification approach has been

applied to various topics in microeconomics, such as labour (Lee, 2009; Gonzalez, 2005; Blundell et al.,

2007), crime (Manski, 2013; Manski and Nagin, 1998), education (Ginther, 2000; Huber et al., 2017) and

health (Gerfin and Schellhorn, 2006; Gundersen et al., 2017, 2012; Kreider et al., 2012).

The identification assumptions employed in the existing literature have not been sufficiently justified

or assessed. In this study, we first re-estimate the causal impacts of the JSY with the DLHS-4, closely

following the approaches taken by Rahman and Pallikadavath (2018), Lim et al. (2010), and Sengupta

and Sinha (2018), which all rely on the CIA. Then we estimate the bounds of ATE through the partial

identification approach. We compare the point-identified impacts with the bound-identified impacts in

order to assess the validity of the assumptions used by their respective studies.

3 Data

3.1 District Level Household and Facility Survey (DLHS)

We use the latest wave of the District Level Household and Facility Survey (DLHS-4), which was conducted

in 2013-2014, where the JSY had been rolled out across the country and had been matured (Rahman

and Pallikadavath, 2017, 2018). Hence the DLHS-4 does not suffer the data contamination problem that

is found in DLHS-3 (Das et al., 2011). The DLHS-4 covers the 18 HPSs5 and three high-performing

union states.6 The DLHS-4 is a representative survey only at the district level, and it collects the data

only in HPSs. So this paper only focuses on the HPSs. We complement our finding in HPSs by proving

additional evidence in LPSs using another latest survey data, the National Family Health Survey (NFHS-

4), in which data have been collected also in LPSs though sample sizes are much smaller. Results for

LPSs are discussed in section A.1 in Appendix A.

3.2 Outcomes

This study estimates the impacts on the following eight maternal and child healthcare utilisations. First,

we estimate the impact on (1) giving birth at health institutions7, which is the primary target outcome

5They are Andhra Pradesh, Arunachal Pradesh, Goa, Haryana, Himachal, Pradesh, Karnataka, Kerala, Maharashtra,Manipur, Meghalaya, Mizoram, Nagaland, Punjab, Sikkim, Tamil Nadu, Telangana, Tripura and West Bengal.

6They are Andaman and Nicobar Islands, Chandigarh, and Puducherry.7Health institution herein includes both public and private health sectors.

9

of the JSY programme and (2) skilled birth attendance. We also estimate the impacts on the uses of (3)

antenatal care (ANC) at least once, (4) ANC three times or more, (5) postnatal care (PNC) for mothers,

(6) PNC for babies, (7) iron and folic acid (IFA) tablets/syrup during pregnancy and (8) tetanus toxoid

(TT) injections to prevent babies from getting tetanus after birth.

3.3 Covariates for the point-identification

In order to make our results comparable with those in the existing literature as much as possible, we control

for the individual- and household-level confounding factors, closely following Rahman and Pallikadavath

(2018). As factors reflecting demographic and socio-economic characteristics, we control for below-the-

poverty-line card ownership, maternal age, a residential location (urban/rural), and a birth order (parity).

A Hindu dummy variable is used to reflect individual socio-cultural backgrounds. Other religions (e.g.,

Muslim, Christianity, Sikhism, Buddhism) are benchmark groups. In addition, we use dummy variables

for the scheduled castes and tribes. Scheduled castes/tribes are the most socially disadvantaged groups,

members of which have suffered the greatest burden of social and economic segregation and deprivation

(Chitnis, 1997). We control for parental educational levels that are measured by the number of education

years. The household wealth is captured through a composite index of relative standards of living.8

3.4 Sample selection

In this study, we focus only on women aged between 15 and 49 who gave birth in the five years prior to the

survey, so that all of those analysed in this study gave birth after the proper implementation of the JSY.

We restrict our attention to the most recent birth of each woman and exclude women who participated

in programmes for childbirth other than the JSY. The DLHS-4 has 84,266 observations. After dropping

observations with missing information9, our sample size has become 67,595. Descriptive statistics are

shown in Table 2.

8Following Rahman and Pallikadavath (2018), we derive the wealth index by applying the principal component analysisover various household characteristics. They are cooking fuel, house type, number of dwelling rooms, electricity, houseownership, landholding, radio, television, computer, internet, telephone, mobile phone, washing machine, refrigerator, sewingmachine, watch, bicycle, motorcycle, car, tractor, tube well, cart, and air cooler.

9We dropped 27 observations because of the missing information about the below-the-poverty-line card ownership. Wedropped 82 and two observations that do not have information about the caste and religions respectively. 2,916 observationshave been dropped because of the missing information about parity. We dropped 79 observations and 13,171 observationsbecause they do not have information about maternal and parental educational backgrounds. We dropped eight observationswith missing information about household wealth. Lastly, we dropped 386 observations with missing information aboutoutcomes.

10

Table 2: Descriptive statistics of the DLHS-4

Non-participants Participantsmean mean

Institutional delivery 0.82 0.95Skilled birth attendance 0.89 0.97Antenatal care at least once 0.87 0.96Antenatal care (≥ 3 times) 0.70 0.81Postnatal care for mother 0.64 0.72Postnatal care for baby 0.78 0.83Iron and folic acid (IFA) supplement 0.68 0.82Tetanus toxoid (TT) injection 0.83 0.93Below the poverty line card 0.30 0.46Scheduled caste 0.77 0.75Scheduled tribe 0.16 0.19Rural 0.56 0.67Birth order (parity) 2.01 1.78Hindu 0.67 0.69Maternal age 27.23 26.05Maternal education years 9.81 9.23Paternal education years 10.92 9.98Wealth 0.54 -0.26

Observations 52732 14863Source: DLHS-4.

4 Methods

4.1 Notations

Let Y = {0, 1} be an indicator for observed health care utilisation. Y becomes 1 if a woman uses a

health care service. D = {0, 1} is an indicator for the participation in the JSY. It equals 1 for those

who participated in the JSY programme and 0 otherwise. Following Rubin (1974), we assume that each

individual has two potential outcomes, namely Y0 in the absence of the treatment and Y1 in the presence

of the treatment. They are latent, in the sense that we can only observe one of them for each person, but

never both. What is actually observed for researchers is

Y = DY1 + (1−D)Y0. (1)

The causal impact of the JSY for individual j ∈ J is Y1j − Y0j , and we are interested in the average

treatment effect (ATE), which is defined by ATE ≡ P (Y1 = 1) − P (Y0 = 1). Theoretically, the ATE

ranges from -1 to +1 and has a width of 2. Let X be observed individual characteristics, which determine

Y and D.

11

Using the law of total probability, we can express P (Y1 = 1) and P (Y0 = 1) as

P (Y1 = 1) = P (Y1 = 1|D = 1)P (D = 1) + P (Y1 = 1|D = 0)︸ ︷︷ ︸Unobservable

P (D = 0) (2)

P (Y0 = 1) = P (Y0 = 1|D = 1)︸ ︷︷ ︸Unobservable

P (D = 1) + P (Y0 = 1|D = 0)P (D = 0), (3)

where P (Y1 = 1|D = 0) is the counterfactual probability that untreated women would use health care

service under the treatment, and P (Y0 = 1|D = 1) is the counterfactual probability that treated women

would use health care service if they had not been treated. P (Y1 = 1|D = 1), P (Y0 = 1|D = 0),

P (D = 1) and P (D = 0) are immediately identifiable from the distribution of the observed data, but

P (Y1 = 1|D = 0) and P (Y0 = 1|D = 1) are unobservable, which makes the ATE unidentifiable without

further assumptions.

4.2 Point-identification with the independence assumption

Researchers must impose assumptions regarding P (Y1 = 1|D = 0) and P (Y0 = 1|D = 1) in order to

identify the ATE. For example, if we assume that participation in the treatment is random, we can point-

identify the ATE from the observed distribution. This assumption is called the independence assumption

and can be expressed as

P (Y1 = 1|D = 0) = P (Y1 = 1|D = 1) (4)

P (Y0 = 1|D = 0) = P (Y0 = 1|D = 1). (5)

The ATE is point-identified by ATE = P (Y1 = 1|D = 1)−P (Y0 = 1|D = 0). This independence assump-

tion is valid under the rigorous RCTs, without which it is usually too stringent. Another identification

assumption that is conventionally imposed in the policy evaluation literature is the conditional indepen-

dence assumption (CIA) in which participation in the treatment is assumed to be random, conditional on

the observable characteristics X. The CIA is formalised by

P (Y1 = 1|D = 0, X) = P (Y1 = 1|D = 1, X) (6)

P (Y0 = 1|D = 1, X) = P (Y0 = 1|D = 1, X). (7)

12

Table 3: Point-identification approaches and assumptionsMethod Short name Assumption

Mean comparison Independence Exogenous/independent treatment assignment

Multivariate OLS Conditional independence;linear regression Outcome linear functional-from;

Additive linearity of the treatment status and the unobservable;Same treatment effect for all individuals

Propensity PSM Conditional independence;score matching Propensity score functional-form

Nearest neighbour matching NNM Conditional independence

OLS adjusted with the IPW+OLS Conditional independenceinverse probability weighting

Entropy balance weighting EBW Conditional independence

Regression discontinuity design RDD Exogenous discontinuity in the probability of the JSY participationat birth order two

For example, evaluations with the multivariate regression approach and the propensity score matching

approach are based on this assumption (Imbens and Wooldridge, 2009; Abadie and Cattaneo, 2018). Al-

though the CIA itself cannot be tested by the data, this assumption can be untenable in many cases. If

the decision to participate in the programme is dependent on unobservable characteristics that can affect

outcomes, the CIA fails, resulting in a biased causal effect estimate.

In this study, we point-estimate the ATEs by (1) mean comparison between the participants and non-

participants, which we call Independence, (2) multivariate or ordinary least squares regression (OLS),

(3) propensity score matching (PSM), (4) OLS regression with inverse probability weighting adjustment

(IPW+OLS), (5) nearest neighbourhood matching (NNM), (6) entropy balance weighting (EBW) and (7)

fuzzy regression discontinuity design (RDD). Their key assumptions are summarised in Table 3.

The mean comparison between the participants and non-participants rests on the independence assump-

tion, and the OLS, PSM, IPW+OLS, NNM and EBW approach rest on the CIA. Some also rely on

functional form assumptions. For example, the OLS assumes that the treatment variable and the error

term are additively separated and the treatment effect is identical for everyone. The PSM and IPW+OLS

rely on the functional form assumption of the propensity score. The EBW non-parametrically balances

the moments of the covariate distributions, the algorithm of which is explained in the Appendix B.1.

13

For fuzzy RDD, we also closely follow the approach taken by Rahman and Pallikadavath (2018), which

exploits the JSY eligibility, i.e. mothers are eligible for the JSY up to their second live birth. The eli-

gibility change of JSY at birth order two was exploited as a source of identification. Following Rahman

and Pallikadavath (2018), we first run OLS regression of the JSY dummy variable on z1 = 1{parity ≤ 2},

z2 = z1 ∗ (parity − 2), z3 = (parity − 2) and the other covariates as the first stage regression, where z1

and z2 are excluded instruments. Using the predicted value of the JSY dummy variable, we estimate the

causal impact of JSY in the second stage OLS regression.

The DLHS is repeated cross-sectional data. If the longitudinal data were available, we could possibly

employ the fixed effect (FE) model, which can deal with the situation where individual unobservable and

time-invariant characteristics influence the selection of the treatment. However, the FE model is less

appealing for the study on maternal health care, because it is not realistic to assume that the degree of

risk-aversion is time-invariable; it can change as mothers experience more childbearing. For example, the

motivation to join the JSY programme for the primiparity can be different from the one for the second or

third childbirth. In this paper, we do not consider the individual-level DID approach either, mainly for

two reasons. First, the JSY is the nationwide programme, which makes it difficult to set up the control

group. Second, we have gaps of more than 10 years between the data collection before and after the JSY

implementation; the DLHS-2 was conducted in 2002-2004. The long periods between the waves could

substantially deteriorate the credibility of common-trend assumption that is essential for the DID.

4.3 Partial identification assumptions

In essence, partial identification first estimates sharp bounds for P (Y1 = 1) and P (Y0 = 1) and then

constructs a sharp bound of ATE. A sharp bound is defined as the narrowest bound that can be obtained

under the maintained assumptions regarding the unobservable distribution. When the bound of P (Yt = 1)

is (LBt, UBt), the ATE bound is defined as

(LB1 − UB0) ≤ ATE ≤ (UB1 − LB0). (8)

Note that first the ATE bound is conceptually different from the confidence interval of the point-estimated

ATE. The width of ATE bound reflects the identification power of the imposed assumptions. The bound

width indicates the tension between the strength of assumptions and their credibility (Manski, 2007),

14

and the bound width does not change with the change in sample sizes. On the other hand, the width

of confidence interval for the point-identified ATE reflects the uncertainty of sampling variability, which

does vary with the change in sample sizes. We discuss the uncertainty of sampling variability for the ATE

bound in section 4.5. In this paper, unless otherwise indicated, we use the term ‘bound’ to denote an

ATE bound of the partial identification and use the term ‘interval’ to indicate a confidence interval.

Second, as the partial identification approach does not assume that participation in the treatment is

random conditional on the covariates, there is no specific need to condition on a long list of covariates.

Hence the partial identification approach is not susceptible to criticisms of omitted variable bias (Manski

and Nagin, 1998). Splitting the sample by a covariate to explore heterogeneity across the population is

also possible for the partial identification as well, but in this paper we do not implement sample splitting.

4.3.1 No assumptions (Worst-case)

First, as a benchmark, we specify a range of P (Y1 = 1|D = 0) and P (Y0 = 1|D = 1) to construct a bound

for P (Y1 = 1) and P (Y0 = 1) without imposing any assumption regarding the counterfactual probabilities.

Since P (Y1 = 1|D = 0) and P (Y0 = 1|D = 1) are probabilities, they necessarily belong to [0,1]. Hence

the bounds of P (Y1 = 1) and P (Y0 = 1) are given from equations (2) and (3) by

P (Y1 = 1|D = 1)P (D = 1)︸ ︷︷ ︸LB1

≤ P (Y1 = 1) ≤ P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)︸ ︷︷ ︸UB1

(9)

P (Y0 = 1|D = 0)P (D = 0)︸ ︷︷ ︸LB0

≤ P (Y0 = 1) ≤ P (Y0 = 1|D = 0)P (D = 0) + P (D = 1)︸ ︷︷ ︸UB0

. (10)

In equation (9), the lower bound of P (Y1 = 1) is attained if all non-participants would have used the

healthcare had they participated in the JSY. The upper bound of P (Y1 = 1) is attained if all non-

participants would have had no health care had they participated in the JSY. The bound width of

P (Y1 = 1) is P (D = 0). The bounds of P (Y0 = 1) in equation (10) can be interpreted in the same fashion

and its bound width is P (D = 1). The sharp bound of ATE can be obtained via equation (8) (Manski,

15

Table 4: Partial-identification assumptions

Assumption Implication

Monotone treatment Individuals do not participate in the programme that makes themresponse (MTR) worse-off with respect to the outcome.

Positive/negative The treated are more/less likely to have the healthcaremonotone treatment than the non-treated both in the presence andselection (MTS) absence of the treatment.

Monotone instrumental Eligible people are less likely to have healthcare in the presence andvariable (MIV) absence of the treatment conditional on the treatment.

1990):

P (Y1 = 1|D = 1)P (D = 1)︸ ︷︷ ︸LB1

−{P (Y0 = 1|D = 0)P (D = 0) + P (D = 1)}︸ ︷︷ ︸UB0

≤ ATE (11)

≤ {P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)}︸ ︷︷ ︸UB1

−P (Y0 = 1|D = 0)P (D = 0)︸ ︷︷ ︸LB0

.

This bound is the narrowest sharp band that can be inferred from the data alone, which is often called a

worst-case bound. Note that even without any assumption about the unobserved probabilities, the data

alone tighten the theoretical width of ATE to half if the support of the outcome variable is bounded

(Manski, 1989). It is known that without any identification assumption, the bound on the ATE in equa-

tion (11) always has a width of 1 and includes 0. Hence without additional assumptions, we cannot

evaluate the sign of the treatment effect on healthcare use. We consider three assumptions: (i) Monotone

treatment response (MTR), (ii) Monotone treatment selection (MTS) and (iii) Monotone instrumental

variable (MIV). The key implications of each assumption is summarised in Table 4.

4.3.2 Monotone treatment response

Second, we assume that individuals do not select a treatment that would make them worse off with respect

to maternal and child healthcare use. This assumption is called the monotone treatment response (MTR)

assumption and it implies that, ceteris paribus, response varies monotonically with treatment (Manski,

1997).

16

Formally, the MTR assumes that for all individual j ∈ J , Y0j ≤ Y1j . We obtain the new bounds for

P (Y1 = 1|D = 0) and P (Y0 = 1|D = 1) as follows:

P (Y0 = 1|D = 0) ≤ P (Y1 = 1|D = 0) ≤ 1 (12)

0 ≤ P (Y0 = 1|D = 1) ≤ P (Y1 = 1|D = 1). (13)

We observe the shrink in the counterfactual probability bounds. For P (Y1 = 1|D = 0), while the bound

of this counterfactual probability in the worst-case ranges from 0 to 1, its new range under the MTR

assumption ranges from P (Y0 = 1|D = 0) to 1. Also, the new range of P (Y0 = 1|D = 1) under the MTR

assumption is from 0 to P (Y1 = 1|D = 1). The following bounds of P (Y1 = 1) and P (Y0 = 1) can be

obtained from equations (1) and (3):

P (Y1 = 1|D = 1)P (D = 1) + P (Y0 = 1|D = 0)P (D = 0)

= P (Y = 1)︸ ︷︷ ︸Updated LB1

≤ P (Y1 = 1) (14)

≤ P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)

and

P (Y0 = 1|D = 0)P (D = 0)

≤ P (Y0 = 1) (15)

≤ P (Y1 = 1|D = 1)P (D = 1) + P (Y0 = 1|D = 0)P (D = 0)

= P (Y = 1)︸ ︷︷ ︸Updated UB0

,

where the lower bound of P (Y1 = 1) and the upper bound of P (Y0 = 1) in the worst-case are replaced

both by P (Y = 1). By equation (8), it follows

0 ≤ ATE ≤ {P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)} − P (Y0 = 1|D = 0)P (D = 0). (16)

Under the MTR assumption, the lower bound of ATE becomes 0. Hence, the MTR assumption pre-

cludes a non-negative ATE and it assumes away the possibility of the detrimental impact of the JSY. In

17

other words, the MTR assumes that the JSY programme would never decrease the likelihood of receiving

healthcare. The MTR assumption, however, leaves open the question of whether the programme has a

strong beneficial effect, mild beneficial effect, or no effect (Gundersen et al., 2017).

In general, the credibility of the MTR assumption depends on the type of policy that we are analysing.

As the MTR assumes the sign of the treatment effect, the MTR assumption can be admittedly too strin-

gent in the case where we have no clue at all regarding the sign of the ATE a priori. However, in the

context of the JSY programme, it is unlikely that a mother does not use the healthcare service when she is

treated and does use the service when she is untreated, because the accredited social health activists, who

support pregnant mothers in the JSY programme, encourage women to receive timely maternal and child

healthcare. Furthermore, they encourage women to undergo three antenatal cares and to give birth in a

health institution. They also assist pregnant women to obtain tetanus toxoid injections and iron/folic acid

supplements. The Ministry of Health and Family Welfare also states that all pregnant mothers should

receive at least three ANCs in their guidelines for the implementation of JSY. Therefore, it is reasonable

to presume that the JSY programme will not decrease healthcare use.

4.3.3 Monotone treatment selection

Third, we make an assumption on the selection mechanism through which women participate in the JSY.

Specifically, we make an assumption as to whether individuals participating in the JSY programme are

more likely to use maternal and child healthcare services on average or not, conditional on treatment

assignment. This assumption is called the monotone treatment selection (MTS) (Manski and Pepper,

2000). The MTS assumption sounds similar to the MTR assumption, but they are different, although

they are not mutually exclusive. In contrast to the MTR assumption, which is on individual-level potential

behaviours, the MTS assumption is on the expected probability. The MTS can be regarded as the weaker

version of the independence assumption, where the equalities in equations (4) and (5) are weakened to

inequalities. There are two types of MTS assumption, namely positive MTS and negative MTS. They

make different assumptions regarding the direction of the selection bias.

(i) Positive MTS: When we assume that mothers participating in the JSY are likely to use no less

healthcare services on average than non-participants, conditional on treatment assignment, this assump-

tion is called the positive MTS assumption. This assumption is plausible in the situations where risk-averse

18

women are more likely to participate in the JSY and use healthcare services than women with lower de-

grees thereof and/or where women in socially and economically segregated ethnic groups know less about

the JSY. The positive MTS assumption is formalised as follows:

0 ≤ P (Y1 = 1|D = 0) ≤ P (Y1 = 1|D = 1) (17)

P (Y0 = 1|D = 0) ≤ P (Y0 = 1|D = 1) ≤ 1. (18)

Note that in equations (17) and (18), the MTS assumes only the direction of the selection bias, and it

does not assume its strength. The positive MTS assumption lowers the upper bound of P (Y1 = 1|D = 0)

in the worst-case from 1 to P (Y1 = 1|D = 1) and raises the lower bound of P (Y0 = 1|D = 1) in the

worst-case from 0 to P (Y0 = 1|D = 0).

Then the new sharp bounds of P (Y1 = 1) and P (Y0 = 1) can be obtained from equations (2) and

(3) as follows (Manski and Pepper, 2000),

P (Y1 = 1|D = 1)P (D = 1) ≤ P (Y1 = 1) ≤ P (Y1 = 1|D = 1)︸ ︷︷ ︸Updated UB1

(19)

P (Y0 = 1|D = 0)︸ ︷︷ ︸Updated LB0

≤ P (Y0 = 1) ≤ P (Y0 = 1|D = 0)P (D = 0) + P (D = 1). (20)

Compared with their bounds in the worst-case, the upper bound of P (Y1 = 1) has become smaller and

the lower bound of P (Y0 = 1) has become larger. The sharp bound of ATE is given via the equation (8)

and it is

P (Y1 = 1|D = 1)P (D = 1)− {P (Y0 = 1|D = 0)P (D = 0) + P (D = 1)}

≤ ATE (21)

≤ P (Y1 = 1|D = 1)− P (Y0 = 1|D = 0).

We find that, compared with the worst-case ATE bound in equation (11), the upper bound in equation

(21) has become smaller. It is easy to show that the upper bound of ATE under the MTS assumption

corresponds to the ATE estimated under the independence assumption (see the note in the Appendix B.2),

which implies that the upper bound of ATE is achieved when there exists no selection bias regarding the

participation in the JSY.

19

(ii) Negative MTS: If we anticipate that mothers who had the greater difficulty in using healthcare

services have higher motivation to participate in the JSY or they are more strongly encouraged by the

accredited social health activists to participate in the JSY, it may be more appropriate to assume that

mothers participating in the JSY are likely to use no more healthcare services on average than non-

participants. This assumption is called the negative MTS assumption. The negative MTS assumption is

formalised as follows:

P (Y1 = 1|D = 1) ≤ P (Y1 = 1|D = 0) ≤ 1 (22)

0 ≤ P (Y0 = 1|D = 1) ≤ P (Y0 = 1|D = 0). (23)

The negative MTS assumption raises the worst-case upper bound of P (Y1 = 1|D = 0) from 0 to P (Y1 =

1|D = 1) and lowers the worst-case lower bound of P (Y0 = 1|D = 1) from 1 to P (Y0 = 1|D = 0). The

bounds of P (Y1 = 1) and P (Y0 = 1) become

P (Y1 = 1|D = 1)︸ ︷︷ ︸Updated LB1

≤ P (Y1 = 1) ≤ P (Y1 = 1|D = 1)P (D = 1) + P (D = 0) (24)

P (Y0 = 1|D = 0)P (D = 0) ≤ P (Y0 = 1) ≤ P (Y0 = 1|D = 0)︸ ︷︷ ︸Updated UB0

. (25)

The sharp bound of ATE is given via the equation (8) and it is

P (Y1 = 1|D = 1)− P (Y0 = 1|D = 0)

≤ ATE (26)

≤ {P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)} − P (Y0 = 1|D = 0)P (D = 0).

We find that, compared with the worst-case bound in equation (11), the lower bound of the ATE in

equation (26) has become larger. Also, this bound shows that the lower bound of ATE under the negative

MTS assumption corresponds to the ATE estimated under the independence assumption, which implies

that the lower bound of ATE is achieved only when there exists no selection bias regarding the partici-

pation in the JSY (see the note in the Appendix B.2).

In general sources of selection bias are multiple. The ultimate direction of the selection mechanism

is expected to vary across the outcomes. Moreover, in essence, there is no definitive way to examine the

20

sources of selection bias and the direction of the bias. However, we can obtain a clue by comparing the

size of ATE estimated under the independence assumption and that estimated under the CIA. When the

ATE under the independence assumption is larger than that under the CIA, it implies the existence of

the positive selection bias, part of which is removed by controlling for covariates. On the other hand,

when the difference of the ATEs is negative, it suggests the existence of the negative selection bias. We

infer the sign of selection bias for each outcome, and impose either positive or negative MTS assumption

accordingly.

4.3.4 Monotone instrumental variable

The last assumption we consider is the monotone instrumental variable (MIV) assumption (Manski and

Pepper, 2000), which states that the latent probability of having healthcare varies weakly and mono-

tonically with an observed instrument, v. Different from the standard instrumental variable approach

(Imbens and Angrist, 1994), the MIV assumption does not impose the mean independence assumption

requiring that the latent outcomes are mean independent of the instrument, namely P (Yt = 1|v = 1) =

P (Yt = 1|v = 0). The instrument v is allowed to be dependent on the mean of potential outcome as

long as the direction of its effect is monotone. Hence, in this sense, the MIV is weaker than the mean

independence assumption. Also the MIV assumption itself does not impose any assumption regarding

the association between the instrument and the treatment status. Thus, the MIV assumption is not

susceptible to criticisms of weak instruments.

Following Gundersen et al. (2017, 2012)10, we use the non-eligibility status of the JSY as an instru-

ment. We assume that ineligible women, who tend to be richer, are more likely to use healthcare services

conditional on the treatment than eligible women. This assumption is supported by the observations

in developing countries that poorer women are less likely to use maternal healthcare services (Pathak

et al., 2010; Pathak and Mohanty, 2010; Kesterton et al., 2010; Balarajan et al., 2011). We discuss the

plausibility of our MIV assumption further in section 6.2.

Formally, let v be a binary instrument, and v = 1 indicates that a woman is not eligible for the JSY. The

10Gundersen et al. (2017, 2012) estimate the ATE bounds of the food assistance programmes targeting poor householdsin the US on nutritional status.

21

MIV assumption imposed herein is expressed as

P (Y1 = 1|v = 0) ≤ P (Y1 = 1|v = 1) (27)

P (Y0 = 1|v = 0)︸ ︷︷ ︸Eligible women

≤ P (Y0 = 1|v = 1)︸ ︷︷ ︸Ineligible women

. (28)

Note that equations (27) and (28) are the assumptions on the latent outcomes. The MTS assumption is a

special case of the MIV assumption, in which the participation status itself is being used an instrument,

i.e v = D.

The MIV assumption gives the following sharp bounds of P (Y1 = 1) and P (Y0 = 1) in equations (29) and

(30). Their derivations are provided in the Appendix B.3.

P (v = 0)P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0)

+P (v = 1) max{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0), P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1)}

≤ P (Y1 = 1) (29)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0) + P (D = 0|v = 0),

P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}

+P (v = 1)[P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)]

and

P (v = 0)P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0)

+P (v = 1) max{P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0), P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1)}

≤ P (Y0 = 1) (30)

≤ P (v = 0) min{P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0) + P (D = 1|v = 0),

P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1) + P (D = 1|v = 1)}

+P (v = 1)[P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1) + P (D = 1|v = 1)].

22

The sharp bound of the ATE can be obtained by equation (8).11 The ATE bounds under the MIV

assumption and those in the worst-case scenario coincide if the worst-case lower and upper bounds of

P (Yt = 1|v = u) weakly increase with u; in such cases, the MIV assumption has no identifying power and

the MIV assumption does not make the ATE bounds narrower (Manski and Pepper, 2000; Richey, 2016).

4.4 Joint imposition of partial identification assumptions

We consider the case where we jointly impose the assumptions introduced so far. We will add mild as-

sumptions one by one and see how much each additional assumption can make the ATE bound narrower.

In the context of the JSY programme, the MTR assumption is the most plausible, followed by the MTS

assumption. Lastly we add the MIV assumption which seems the most strongest of all three assumptions.

Hence, in this paper we mainly consider the following two cases: MTR+MTS and MTR+MTS+MIV.

When we jointly impose different assumptions, we derive the bounds of P (Y1 = 1) and P (Y0 = 1)

under the jointly imposed assumptions in order to get the ATE bound. ATE bounds estimators and their

full derivations are in the Appendix C. For the other combinations, i.e, MTR+MIV and MTS+MIV,

their estimators are also provided in the Appendix D.

4.5 Inference of partial identification

Estimating the ATE bounds requires us to consider sampling variabilities. The ATE bounds introduced

so far concern the conclusions that could be drawn under different assumptions if we could observe the

JSY participation status and outcomes experienced by everyone in the population. Sampling variability,

however, arises when these data are available for only a sample of the population. We consider the sample

variability by constructing the confidence intervals for the bound estimates.

A confidence interval for the bound estimate is the area that contains the parameter of interest. However,

11It is known that when we estimate the ATE bound under the MIV assumption with the sample analogue, the estimatecan suffer finite-sample bias which could potentially lead the bound to be narrower than the true bound (Manski andPepper, 2009). By Jensen’s inequality, the estimated lower bounds are potentially biased upwards because of their maximaoperators, and the estimated upper bounds are potentially biased downwards because of their minima operators. Detailsare provided in the Appendix B.4. To address this bias, we implement a bootstrap-based correction proposed by Kreiderand Pepper (2007). This method estimates the bias by using the bootstrap distribution and adjusts the sample analogueestimate in accordance with the estimated bias. For example, when we have a random sample of size N and let LBN

be the sample analogue estimate of the lower bound in question. We denote Eb(LBN ) as the mean of the estimate fromthe bootstrap distribution of size b. The bias is estimated as Eb(LBN ) − LBN . The bias-corrected estimate is given byLBN − {Eb(LBN ) − LBN} = 2LBN − Eb(LBN ). We correct the bias of the upper bound in the same way. In this paperwe estimate the bias with b = 100 bootstrap repetitions. The performance of this method is confirmed by Monte Carlosimulations in Manski and Pepper (2009).

23

there is no consensus on its definition and its research is still ongoing. When we consider a confidence

interval in the partial identification, the question arises as to whether to construct a interval over the

region of identification (Chernozhukov et al., 2007) or over the actual parameter of interest (Imbens and

Manski, 2004).12

We obtain the confidence intervals for the estimated bounds by the method developed by Imbens and

Manski (2004). The confidence intervals defined by Imbens and Manski (2004) are the ones that asymp-

totically cover the true parameter, namely ATE, with a fixed probability, rather than covering the entire

identification region with fixed probability.

Formally, the confidence intervals for parameter θ at the 1 − α level are defined as the sets of the pa-

rameters where we cannot reject the null hypothesis that θ is an element of the identification set, Θ, at

the α level. When the estimated lower and upper bounds are [LB, UB] and their standard errors are

σLB and σUB, 1− α level confidence intervals of Imbens and Manski (2004) are constructed as CI1−α ∈

[LB−cσLB, UB+cσUB], where the parameter c is the one that solves Φ(c+ (σUB−σLB)max{σLB ,σUB})−Φ(−c) = 1−α

with the Newton-Raphson method. For more details, see Imbens and Manski (2004). The confidence in-

tervals are obtained by bootstrap with 200 repetitions.

5 Results

5.1 Point-identification

We point-estimate the ATEs by: (1) Independence, (2) OLS, (3) PSM, (4) IPW+OLS, (5) NNM, (6)

EBW and (7) RDD. For these approaches (2)-(7), we use the set of covariates used in Rahman and Pal-

likadavath (2018). The logistic regression is used to estimate the propensity score.13 Table 5 shows the

estimation results, where we observe that the JSY has positive impacts on all outcomes (p < 0.01). When

we relax the independence assumption and instead impose the CIA, we find larger effects on institutional

delivery, skilled birth attendance, PNC for mothers and PNC for babies, and smaller effects on ANC at

least once, ANC at least three times, iron and folic acid supplement intakes and tetanus toxoid injections.

12Instead of them, some studies provide confidence internals for the upper and lower ATE bounds respectively (Ginther,2000).

13Its result is available from the author upon request.

24

The results of the fuzzy RDD approach are shown in the last two columns of Table 5. They show

the largest ATEs on all outcomes except PNC for mothers. As is well known, the RDD estimates the

ATE among the sub-population that are induced to participate in the treatment with the change in instru-

mental variables, and these people are often called “compliers”. Hence, an ATE estimated by RDD can

be different from ATEs estimated by other approaches if compliers are very much different from the entire

sample population (Imbens and Angrist, 1994). The jumps and kinks in the probability of participating

in the programme and the probability of using the healthcare are shown in Figure 2.

Figure 2: Discontinuity at parity more than two

Source: DLHS-4. Note: ANC=Antenatal care; PNC=Postnatal care; IFA=Iron and folic acid; and TT=tetanus toxoid.

5.2 Partial identification

Table 6 is the main result of this study, which reports the change in the bound estimates for each

outcome as we incrementally tighten the ATE bounds by adding stronger assumptions one by one. Herein

we are considering the following four cases: (1) No assumption, (2) MTR, (3) MTR+MTS, and (4)

25

Table 5: Point-estimated ATEEstimates SEs Confidence intervals

Institutional deliveryIndependence 0.126*** 0.003 (0.12, 0.132)OLS 0.129*** 0.003 (0.123, 0.135)PSM 0.129*** 0.003 (0.123, 0.135)IPW+OLS 0.132*** 0.003 (0.126, 0.138)NNM 0.122*** 0.003 (0.116, 0.128)EBW 0.131*** 0.003 (0.125, 0.137)RDD 0.19*** 0.037 (0.117, 0.263)

Skilled delivery attendanceIndependence 0.081*** 0.002 (0.077, 0.085)OLS 0.088*** 0.002 (0.084, 0.092)PSM 0.088*** 0.002 (0.084, 0.092)IPW+OLS 0.09*** 0.002 (0.086, 0.094)NNM 0.082*** 0.002 (0.078, 0.086)EBW 0.094*** 0.002 (0.09, 0.098)RDD 0.134*** 0.030 (0.075, 0.193)

Antenatal care +1Independence 0.091*** 0.002 (0.087, 0.095)OLS 0.079*** 0.003 (0.073, 0.085)PSM 0.078*** 0.003 (0.072, 0.084)IPW+OLS 0.08*** 0.003 (0.074, 0.086)NNM 0.072*** 0.003 (0.066, 0.078)EBW 0.079*** 0.003 (0.073, 0.085)RDD 0.139*** 0.035 (0.07, 0.208)

Antenatal care +3Independence 0.108*** 0.004 (0.1, 0.116)OLS 0.082*** 0.005 (0.072, 0.092)PSM 0.07*** 0.006 (0.058, 0.082)IPW+OLS 0.079*** 0.005 (0.069, 0.089)NNM 0.075*** 0.006 (0.063, 0.087)EBW 0.091*** 0.004 (0.083, 0.099)RDD 0.129*** 0.050 (0.031, 0.227)

Postnatal care for motherIndependence 0.086*** 0.004 (0.078, 0.094)OLS 0.105*** 0.005 (0.095, 0.115)PSM 0.094*** 0.006 (0.082, 0.106)IPW+OLS 0.106*** 0.005 (0.096, 0.116)NNM 0.102*** 0.006 (0.09, 0.114)EBW 0.106*** 0.004 (0.098, 0.114)RDD 0.071 0.051 (-0.029, 0.171)

Postnatal care for babyIndependence 0.054*** 0.004 (0.046, 0.062)OLS 0.076*** 0.004 (0.068, 0.084)PSM 0.068*** 0.005 (0.058, 0.078)IPW+OLS 0.077*** 0.004 (0.069, 0.085)NNM 0.073*** 0.004 (0.065, 0.081)EBW 0.071*** 0.004 (0.063, 0.079)RDD 0.113*** 0.044 (0.027, 0.199)

Iron folic acid supplement intakesIndependence 0.14*** 0.004 (0.132, 0.148)OLS 0.099*** 0.005 (0.089, 0.109)PSM 0.091*** 0.006 (0.079, 0.103)IPW+OLS 0.096*** 0.005 (0.086, 0.106)NNM 0.085*** 0.006 (0.073, 0.097)EBW 0.105*** 0.004 (0.097, 0.113)RDD 0.145*** 0.051 (0.045, 0.245)

Tetanus toxoid injectionsIndependence 0.099*** 0.003 (0.093, 0.105)OLS 0.088*** 0.003 (0.082, 0.094)PSM 0.088*** 0.004 (0.08, 0.096)IPW+OLS 0.088*** 0.003 (0.082, 0.094)NNM 0.078*** 0.004 (0.07, 0.086)EBW 0.09*** 0.003 (0.084, 0.096)RDD 0.17*** 0.039 (0.094, 0.246)

Source: DLHS-4. Note: Number of observations is 67,595. The choice of covariates follows Rahman and Pallikadavath (2018). 95%confidence intervals are shown. PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverseprobability weighting; NNM=Nearest neighbour matching; and EBW=Entropy balance weighting; and RDD=Regression discontinuitydesign. * p < 0.10, ** p < 0.05, *** p < 0.01.

26

MTR+MTS+MIV. Table 6 also reports the confidence intervals for the respective ATE bound estimates.

We observe that the widths of the confidence intervals are just slightly wider than the widths of the

estimated ATE bounds. Compared with the widths of the bound estimates, the additional widths due

to the sampling variability are very small, which is thanks to the large sample sizes. This suggests that

identification problem is the dominant concern in inference on the impacts and sampling variance poses

a much less serious problem for inference (Manski and Nagin, 1998).

Institutional delivery

First, for institutional delivery, in the worst-case scenario where we do not impose any assumption, the

lower bound is -0.650 and the upper bound is 0.350. In the worst-case, the bound width is always 1 and

the bound always includes 0. This bound just ensures that the true ATE can never be outside of this

bound as long as the ATE is estimated from this data. Although this bound has a wider negative range,

it does not necessarily mean that the ATE is more likely to be negative. Also, the centre of the bound

(-0.15 in this case) is not necessarily the most probable ATE.

When we impose the MTR assumption, it truncates the lower bound of ATE at 0 because the MTR

assumes that the ATE is non-negative. Then we add the MTS assumption to the MTR assumption.

As the ATE estimated by OLS is larger than the ATE estimated by mean comparisons, we assume the

negative MTS assumption here. The negative selection bias assumes that individual observable and un-

observable characteristics are correlated with the participation in the programme and the institutional

delivery in the opposite directions. One possible case would be that richer mothers who seek high quality

healthcare at private hospitals may have lower motivation to participate in the JSY. Imposing the MTS

assumption, we find that the lower bound becomes even larger – from 0 to 0.126. Further, adding the

MIV assumption makes the ATE bound even narrower. The lower bound has become 0.262 and the upper

bound has become 0.331.

After adding the MIV assumption, we find that all point-identified ATE estimates are outside of the

ATE bound. The ATEs on institutional delivery estimated under the CIA are smaller than the lower

ATE bound estimated under the MTR, MTS and MIV assumptions, thereby suggesting that their main-

tained assumptions are not compatible with the MTR, MTS and MIV assumptions used in the partial

27

Table 6: ATE bound estimatesEstimates Confidence intervals

Lower bound Upper bound Lower bound Upper bound

Institutional deliveryNo assumption −0.652 0.348 −0.654 0.351MTR 0.000 0.348 0.000 0.351MTR+MTS 0.126 0.348 0.122 0.351MTR+MTS+MIV 0.262 0.331 0.255 0.335

Skilled delivery attendanceNo assumption −0.702 0.298 −0.705 0.301MTR 0.000 0.298 0.000 0.301MTR+MTS 0.081 0.298 0.078 0.301MTR+MTS+MIV 0.176 0.294 0.170 0.297

Antenatal care +1No assumption −0.685 0.315 −0.688 0.319MTR 0.000 0.315 0.000 0.319MTR+MTS 0.000 0.091 0.000 0.095MTR+MTS+MIV 0.000 0.042 0.000 0.050

Antenatal care +3No assumption −0.587 0.413 −0.590 0.417MTR 0.000 0.413 0.000 0.417MTR+MTS 0.000 0.108 0.000 0.115MTR+MTS+MIV 0.000 0.014 0.000 0.029

Postnatal care for motherNo assumption −0.558 0.442 −0.561 0.445MTR 0.000 0.442 0.000 0.445MTR+MTS 0.086 0.442 0.078 0.445MTR+MTS+MIV 0.196 0.430 0.187 0.434

Postnatal care for babyNo assumption −0.644 0.356 −0.647 0.358MTR 0.000 0.356 0.000 0.358MTR+MTS 0.054 0.356 0.048 0.358MTR+MTS+MIV 0.158 0.349 0.150 0.352

Iron folic acid supplement intakesNo assumption −0.570 0.430 −0.573 0.433MTR 0.000 0.430 0.000 0.433MTR+MTS 0.000 0.140 0.000 0.146MTR+MTS+MIV 0.000 0.065 0.000 0.081

Tetanus toxoid injectionsNo assumption −0.664 0.336 −0.668 0.339MTR 0.000 0.336 0.000 0.339MTR+MTS 0.000 0.099 0.000 0.104MTR+MTS+MIV 0.000 0.046 0.000 0.056

Source: DLHS-4. Note: Number of observations is 67,595. Note: 95% confidence intervals are calculated following Imbensand Manski (2004) by bootstrap with 200 repetitions.

28

identification approach.14 This incompatibility is because: (1) the CIA is not valid; (2) functional-form

assumptions are not correct; and/or (3) the MTR, MTS, and MIV assumptions are not valid. The finding

that the non-parametric EBW estimate is outside of the bound implies that regardless of functional-form

assumptions, the point-identified ATE is outside of the ATE bound.

Moreover, as we discussed in the previous section, the identification assumptions used for the bound

estimation are supported by empirical and observational evidence. Given that the CIA is hard to be jus-

tified for the case of the JSY, it would be reasonable to think that the incompatibility between assumptions

for the point-identification and those for the partial-identification mainly comes from the possibly unac-

ceptable CIA and/or functional form specification errors.15 Also, we found that the ATE estimated by

the fuzzy RDD is also smaller than the lower bound of ATE, which implies that the compliers are very

much different from the other populations and/or the assumption of the discontinuity is not valid.

Skilled birth attendance

For skilled birth attendance, in the worst-case, the ATE bound is from -0.702 to 0.298. The MTR assump-

tion truncates the lower bound at 0. As the ATE estimated by OLS is larger than the ATE estimated by

mean comparisons, we assume the negative MTS assumption. We speculate that most of the rich mothers

who can afford to give birth in the presence of skilled birth attendants may not have been eligible for the

JYS. Adding the negative MTS assumption raises the lower bound up to 0.081. Further adding the MIV

assumption makes the band narrower. Under the three assumptions, the lower and upper bounds have

become 0.176 and 0.294 respectively (Figure 4).

All point-estimated ATEs are all smaller than the lower limit of this ATE bound, which suggests that

they could be under-estimated under the maintained assumptions. The confidence interval of the ATE

bound under the MTR, MTS and MIV assumptions is between 0.170 and 0.297, which does not include

the point-identified ATEs.

14Strictly speaking, even without the MTR assumption, all the point-identified ATEs are found to be outside of the boundestimates (see Figure 4).

15We estimated the propensity score differently with the probit and linear probability model, but we did obtain similarresults.

29

ANC use at least once

For ANC use at least once, in the worst-case, the ATE bound is from -0.685 to 0.315. The MTR assump-

tion makes the bound non-negative. As the ATE estimated by OLS is smaller than the ATE estimated

by mean comparisons, we add the positive MTS assumption to the MTR. The positive MTS assumption

implies that mothers with higher awareness of the importance of maternal healthcare may be more likely

to participate in the JSY and receive ANC. Adding the MTS assumption substantially narrows the upper

bound of ATE to 0.091, implying that the assumption regarding the direction of selection bias has large

identifying power. The MIV assumption further narrows the upper bound to 0.042.

We observe that the ATEs under the independence assumption and the CIA lie beyond the upper limit

of this ATE bound (Figure 4). We also observe that the ATE estimated by the RDD is outside of this

bound, thereby implying that the compliers are very much different from the entire populations and/or

the assumption of the discontinuity is not valid. We do not observe the overlapping of confidence intervals

between point-estimated ATEs and bound-estimated ATE.

ANC use at least three times

We also observe similar shrinks in the ATE bound for ANC use three times and more. Its worst-case

bound ranges from -0.587 to 0.413. The MTR assumption truncates the lower bound at 0. As the ATE

estimated by OLS is smaller than the ATE estimated by mean comparisons, we assume the positive MTS

assumption. Adding the positive MTS assumption narrows the upper bound from 0.413 to 0.108. The

MIV assumption further narrows it up to 0.014.

After adding the MIV assumption, the point-identified ATEs are again beyond this upper bound (Fig-

ure 4). We do not observe the overlapping of confidence intervals between point-estimated ATEs and

bound-estimated ATE.

PNC for mothers

For PNC for mothers, the worst-case bound is -0.558 to 0.442 and the MTR assumption makes it non-

negative. As the ATE estimated by OLS is larger than the ATE estimated by mean comparisons, we

assume the negative MTS assumption. We speculate that the observed negative selection bias suggests

that those mothers who have greater difficulty in having healthcare may have been more strongly en-

30

couraged by the accredited social health activists to join the JSY programme or/and that mothers who

can afford to receive PNC may not have been eligible for the JSY. We find that the negative MTS has

large identifying power and succeeds in increasing the lower bound up to 0.086. The MIV assumption

contributes to further narrowing the bounds of the ATE.

Under the three assumptions, we observe that the ATE is no less than 0.193 and no larger than 0.430.

Under the three assumptions, we observe that the point-identified ATEs are all smaller than the lower

limit of ATE bound, suggesting the possibility of underestimation of ATEs under the CIA (Figure 5).

The ATE under the RDD assumption lies also outside the bound. We do not observe the overlapping of

confidence intervals between point-estimated ATEs and bound-estimated ATE.

PNC for babies

For PNC for babies, the worst-case bound is from -0.644 to 0.356. Along with PNC for mothers, as the

ATE estimated by OLS is larger than the ATE estimated by mean comparisons, we assume the negative

MTS assumption. The MTR and negative MTS assumptions raise the lower limit of the ATE bound up to

0.054. The MIV assumption further narrows the lower bound to 0.158 and the upper bound to 0.349. We

find that the ATEs under the CIA and the RDD assumption lie below this lower bound, suggesting that

they could be under-estimated (Figure 5). Even when considering the sampling variability, the confidence

interval of the ATE bound does not include the point-estimated ATEs.

Iron and folic acid supplement intakes

For iron and folic acid supplement intakes, we find that in the worst-case, the ATE ranges from -0.570

to 0.430. As the ATE estimated by OLS is smaller than the ATE estimated by mean comparisons, we

impose the positive MTS assumption, implying that mothers who understand the importance of maternal

healthcare more are more likely to participate in the programme and receive the supplement. The MTR

assumption makes the bound non-negative and the positive MTS assumption substantially narrows the

upper bound to 0.140. Adding the MIV assumption further narrows the upper bound up to 0.065. We find

that all point-estimated ATEs are greater than the upper bound, thereby suggesting their over-estimation

(Figure 5). Although the confidence interval of the ATE bound does not include the point-identified ATEs,

but we observe the overlapped confidence intervals of the ATEs point-estimated by PSM, NNM and RDD

with that of the bound-estimated ATE.

31

Tetanus toxoid injections

Finally, for the tetanus toxoid injections, the worst-case bound is from -0.664 to 0.336. The MTR

assumption truncates the lower bound at 0. As with the case with iron and folic acid supplement,

as the ATE estimated by OLS is smaller than the ATE estimated by mean comparisons, we assume

the positive MTS assumption. We conjecture that more risk averse mothers are more likely to join the

JSY and get the treatment. We observe that the positive MTS assumption substantially contributes to

narrowing the upper bound up to 0.099, and adding the MIV assumption contributes to further narrowing

it up to 0.046. We find that the point-estimated ATEs are all outside of the bound again, indicating the

evidence of over-estimation (Figure 5). We do not observe the overlapping of confidence intervals between

point-estimated ATEs and bound-estimated ATE.

5.3 Result summary

Figure 3 visually summarises the results presented above. The bounds shown in Figure 3 are the sharp

bounds under the MTR, MTS and MIV assumptions. We find that the point-identified ATEs are below

the lower ATE bound of institutional delivery, skilled birth attendance, PNC for mothers and children.

On the other hand, for ANC at least once and ANC at least three times, the point-identified ATEs

are over the upper limit of the ATE bounds. In particular, a larger deviation from the lower bound is

observed for institutional delivery. Even considering the uncertainty of sampling variability, we do not

observe the overlapping of the confidence intervals between point-identified and partially-identified ATEs

for institutional delivery, ANC at least once, PNC for mothers and tetanus toxoid injections.

Next we explore how each of the three assumptions contributes to estimating the ATE bounds for each

outcome. Figures 4 and 5 illustrate the sharp ATE bounds for each outcome estimated under the eight

different combinations of assumptions. By comparing the bound widths, we can infer the identification

power of each assumption. The joint imposition of the MTS and MIV assumptions leads to our main

finding that point-estimated ATEs are outside the bound-estimated ATEs. Interestingly, even without the

MTR assumption, we could have reached the same finding although the estimated ATE bound can have

a larger width. Another interesting finding is that the MTR assumption is not binding in the presence of

the negative MTS assumption. This is because the negative MTS assumption itself made the lower limits

of the ATE bounds positive. Nevertheless, the MTR assumption significantly contributes to narrowing

the ATE bounds unless the negative MTS assumption is imposed.

32

Figure 3: Point-estimated ATE and bound-estimated ATE under the MTR, MTS and MIV assumptions

Source: DLHS-4. Note: The choice of covariates follows Rahman and Pallikadavath (2018).PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverse probability weighting;NNM=Nearest neighbour matching; EBW=Entropy balance weighting; RDD=Regression discontinuity design;ANC=Antenatal care; PNC=Postnatal care; IFA=Iron and folic acid; and TT=Tetanus toxoid. The bounds shown aresharp bounds estimated under the MTR, MTS and MIV assumptions. The shaded areas show the 95% level confidenceinterval of the ATE bound under the MTR, MTS and MIV assumptions.

33

Figure 4: ATE bounds for institutional delivery, Skilled birth attendance and ANC use

Source: DLHS-4. Note: The choice of covariates follows Rahman and Pallikadavath (2018).PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverse probability weighting;NNM=Nearest neighbour matching; EBW=Entropy balance weighting; and RDD=Regression discontinuity design. Thebounds shown are sharp bounds under the MTR, MTS and MIV assumptions. The shaded areas show the 95% levelconfidence interval of the ATE bound under the MTR, MTS and MIV assumptions.

34

Figure 5: ATE bounds for PNC, iron and folic acid supplement intakes and tetanus toxoid injections

Source: DLHS-4. Note: The choice of covariates follows Rahman and Pallikadavath (2018).PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverse probability weighting;NNM=Nearest neighbour matching; EBW=Entropy balance weighting; and RDD=Regression discontinuity design. Thebounds shown are sharp bounds under the MTR, MTS and MIV assumptions. The shaded areas show the 95% levelconfidence interval of the ATE bound under the MTR, MTS and MIV assumptions.

35

6 Robustness checks and further analysis

6.1 Robustness check: choice of covariates

As robustness checks with respect to the choice of covariates, we point-identify ATEs again with different

covariate sets used in the previous studies and the full covariate sets in order to see whether the choice of

covariates substantially affects our findings. Note that as an ATE bound is unconditional on covariates,

it is invariant across the covariate sets. First, we closely follow Lim et al. (2010) and use the following

covariate sets: state of residence, urban residence, below-the-poverty-line card ownership, wealth quintile,

scheduled caste, education, parity, and maternal age.

Next, we closely follow Sengupta and Sinha (2018), in which the authors used the following variables

as determinants of the JSY participation and healthcare use: residence in North-Western states, urban

residence, below-the-poverty-line card ownership, number of live births, scheduled caste, scheduled tribe,

whether pregnancy was known within three months, any previous stillbirths, any previous miscarriages,

any children who died, age at which living with husband, age at time of birth, ratio of male to female

members in household, being Hindu, wealth quintiles, living in a kachcha (made of natural materials)

house, maternal and paternal education dummies.16 As well as them, following Sengupta and Sinha

(2018), we use the following variables that affect the healthcare use only: sex of baby, number of brothers

and sisters, whether any household member is covered by health insurance, having no toilet facility, and

any water treatment for drinking.

Third, we control for more variables to fully take into account of the selection bias as much as we

could. We control for the following covariates: birth-year fixed effects, birth-month fixed effects, state-

fixed effects, urban residence, below-the-poverty-line card ownership, number of live births, scheduled

caste, scheduled tribe, whether pregnancy was known within three months, any previous stillbirths, any

previous miscarriages, any children who died, age at which living with husband, age at time of birth,

ratio of male to female members in household, wealth quintiles, maternal and paternal education levels

and being Hindu. This complete set of covariates is expected to more effectively remove the selection bias

attributable to measurable characteristics. If the point-estimated ATEs estimated with this full covariate

set are outside of the ATE bounds, then it would support our finding that the CIA would be implausible

16We defined the following three educational levels for mothers and fathers respectively: 1-6 years of education, 7-12 yearsof education, and more than 12 years of education.

36

and the estimated ATEs could be biased.

The results estimated with these three sets of covariates are shown in Table 7 and summarised visually

in Figure 6. We find that all point-identified ATEs lie outside of ATE bound in three sets of covariates.

We reconfirm that the point-identified ATEs are likely to be over-estimated for ANC, iron and folic acid

supplement intakes and tetanus toxoid injections and to be under-estimated for institutional delivery,

skilled birth attendance and PNC. Overall, these findings are consistent with our main results in Figure

3, implying that regardless of the choice of covariate sets, the conditional independence and functional

form assumptions are likely to be invalid.

6.2 Robustness check: validity of the MIV assumption

Manski (2003) argues that the credibility of estimates decreases with the strength of assumptions main-

tained (The Law of Decreasing Credibility). We observe that, in both HPSs and LPSs, the MIV assumption

made a key role in our finding that point-identified ATEs lie outside the bound-estimated ATEs. Hence

it is important to examine the credibility of the MIV assumption. Although a large volume of literature

reports that socio-economic status is negatively associated with maternal and child healthcare use in

developing countries (Pathak et al., 2010; Pathak and Mohanty, 2010; Kesterton et al., 2010; Balarajan

et al., 2011), we cannot directly test the validity of MIV assumption formalised in equations (27) and (28)

in principle, because these are assumptions on latent probabilities. Nevertheless, we can still explore the

validity of equation (27), using the National Family Health Survey data, which was surveyed in 2005-2006.

By focusing only on women who delivered before 12 April 2005, we can observe both P (Y0 = 1|v = 0)

and P (Y0 = 1|v = 1) among women who gave birth before the introduction of the JSY. Figure 7 shows

that P (Y0 = 1|v = 0) is larger than P (Y0 = 1|v = 1) for all outcomes (p < 0.01), supporting the validity

of equation (27) and providing confidence in our results.

7 Conclusion

In India, the CCT programme, the JSY, was introduced in 2005, which promotes the use of maternal and

child healthcare with cash incentives in order to reduce infant and neonatal mortalities. In contrast to

other countries with CCT programmes, a rigorous RCT was not conducted in India for the JSY (Joshi and

Sivaram, 2014), and hence a valid instrumental variable has not been available for researchers. Lim et al.

37

Table 7: Point-estimated ATE with different covariate setsLim et al. (2010) Sengupta et al. (2018) Full setsEstimates SEs Estimates SEs Estimates SEs

Institutional deliveryIndependence 0.432*** 0.005 0.432*** 0.005 0.432*** 0.005OLS 0.127*** 0.003 0.131*** 0.002 0.128*** 0.003PSM 0.124*** 0.003 0.129*** 0.003 0.127*** 0.003IPW+OLS 0.129*** 0.003 0.131*** 0.003 0.131*** 0.003NNM 0.122*** 0.003 0.126*** 0.003 0.12*** 0.003EBW 0.131*** 0.003 0.139*** 0.003 0.13*** 0.003

Skilled delivery attendanceIndependence 0.349*** 0.005 0.349*** 0.005 0.349*** 0.005OLS 0.087*** 0.002 0.086*** 0.002 0.087*** 0.002PSM 0.083*** 0.002 0.087*** 0.002 0.086*** 0.002IPW+OLS 0.088*** 0.002 0.086*** 0.002 0.089*** 0.002NNM 0.081*** 0.002 0.081*** 0.002 0.082*** 0.002EBW 0.094*** 0.002 0.095*** 0.002 0.092*** 0.002

Antenatal care +1Independence 0.105*** 0.005 0.105*** 0.005 0.105*** 0.005OLS 0.078*** 0.003 0.089*** 0.002 0.078*** 0.003PSM 0.075*** 0.004 0.089*** 0.003 0.074*** 0.004IPW+OLS 0.078*** 0.003 0.087*** 0.003 0.079*** 0.003NNM 0.074*** 0.004 0.085*** 0.003 0.075*** 0.004EBW 0.078*** 0.003 0.092*** 0.002 0.076*** 0.003

Antenatal care +3Independence 0.101*** 0.007 0.101*** 0.007 0.101*** 0.007OLS 0.079*** 0.005 0.098*** 0.004 0.079*** 0.005PSM 0.074*** 0.006 0.093*** 0.006 0.075*** 0.006IPW+OLS 0.078*** 0.005 0.093*** 0.004 0.077*** 0.005NNM 0.07*** 0.006 0.091*** 0.005 0.073*** 0.005EBW 0.092*** 0.004 0.109*** 0.004 0.089*** 0.004

Postnatal care for motherIndependence 0.178*** 0.007 0.178*** 0.007 0.178*** 0.007OLS 0.104*** 0.005 0.088*** 0.005 0.104*** 0.005PSM 0.111*** 0.006 0.084*** 0.006 0.097*** 0.006IPW+OLS 0.106*** 0.005 0.089*** 0.005 0.107*** 0.005NNM 0.104*** 0.006 0.09*** 0.006 0.094*** 0.006EBW 0.106*** 0.004 0.094*** 0.004 0.108*** 0.004

Postnatal care for babyIndependence 0.065*** 0.007 0.065*** 0.007 0.065*** 0.007OLS 0.074*** 0.004 0.058*** 0.004 0.075*** 0.004PSM 0.076*** 0.005 0.055*** 0.005 0.068*** 0.005IPW+OLS 0.075*** 0.004 0.059*** 0.004 0.077*** 0.004NNM 0.071*** 0.004 0.062*** 0.004 0.064*** 0.004EBW 0.071*** 0.004 0.06*** 0.004 0.072*** 0.004

Iron folic acid supplement intakesIndependence 0.104*** 0.006 0.104*** 0.006 0.104*** 0.006OLS 0.098*** 0.005 0.125*** 0.004 0.097*** 0.005PSM 0.087*** 0.006 0.126*** 0.005 0.095*** 0.006IPW+OLS 0.096*** 0.005 0.119*** 0.004 0.094*** 0.005NNM 0.086*** 0.006 0.118*** 0.005 0.09*** 0.006EBW 0.105*** 0.004 0.126*** 0.004 0.101*** 0.004

Tetanus toxoid injectionsIndependence 0.056*** 0.004 0.056*** 0.004 0.056*** 0.004OLS 0.086*** 0.003 0.098*** 0.003 0.087*** 0.003PSM 0.084*** 0.004 0.097*** 0.004 0.083*** 0.004IPW+OLS 0.086*** 0.003 0.095*** 0.003 0.087*** 0.004NNM 0.079*** 0.004 0.095*** 0.004 0.083*** 0.004EBW 0.09*** 0.003 0.101*** 0.003 0.087*** 0.003

Source: DLHS-4. Note: 95% confidence intervals are shown. PSM=Propensity score matching; IPW+OLS=OLS adjustedwith the propensity score inverse probability weighting; NNM=Nearest neighbour matching; and EBW=Entropy balanceweighting. * p < 0.10, ** p < 0.05, *** p < 0.01.

38

Figure 6: ATE bound and point-estimated ATE with different covariate sets

Source: DLHS-4. Note: PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverseprobability weighting; NNM=Nearest neighbour matching; EBW=Entropy balance weighting; ANC=Antenatal care;PNC=Postnatal care; IFA=Iron and folic acid; and TT=tetanus toxoid. The bounds shown are sharp bounds estimatedunder the MTR, MTS and MIV assumptions. The shaded areas show the 95% level confidence interval of the ATE boundunder the MTR, MTS and MIV assumptions.

39

Figure 7: Comparison in the mean outcomes between eligible and non-eligible mothers without thetreatment

Source: National Family Health Survey. Note: 95% confidence intervals are shown.

(2010) conducted a first estimate of the causal impacts of the JSY, which has attracted much attention

for years. Up to now, several studies have attempted to estimate its causal impacts on various healthcare

services and health outcomes. However, according to Das et al. (2011), the JSY was immature until 2007

and the data set analysed by Lim et al. (2010) and others (i.e., DLHS-3) was not reliable enough to

estimate the causal impact. The latest wave of the DLHS (DLHS-4) has become available only recently,

and Rahman and Pallikadavath (2017) and Rahman and Pallikadavath (2018) re-estimated the causal

impacts, acknowledging the potential serious errors that had occurred in DLHS-3.

Up until now, however, no study has ever tried to assess the validity of the identification assumptions

employed in previous studies. For example, Lim et al. (2010), Rahman and Pallikadavath (2018) and Sen-

gupta and Sinha (2018) all relied on the CIA that participation in the JSY programme can be regarded

as random if we control for observable household and individual characteristics. However, the CIA is

extremely hard to justify for the case of JSY. The utilisation of maternal and child healthcare use and

the participation in the JSY are likely to be dependent on individual unobservable characteristics such as

the degree of being risk-averse and the awareness of the importance of maternal and child healthcare. If

40

there exist some selection mechanisms that cannot be controlled for by the observable characteristics, the

estimated ATE under the CIA suffers selection bias, potentially resulting in flawed and conflicting conclu-

sions (Manski, 2013). This study assessed the validity of the identification assumptions used in previous

studies and provided new evidence through a partial identification approach. The partial identification

taken herein yielded an honest and credible bound of ATE, which is valuable when we are not confident

of the conventionally imposed identification assumptions.

If the imposed assumptions are valid, any point-estimate of ATE should lie within the ATE bound.

However, we find that the ATEs under the CIA lie beyond the ATE bounds in HPSs, suggesting the

invalidity of identification assumptions and the possibility of over- or under-estimation. Especially, we

find that the point-identified ATEs are below the lower ATE bounds of institutional delivery, skilled birth

attendance, PNC for mothers and children. On the other hand, for ANC at least once and ANC at least

three times, the point-identified ATEs are over the upper limits of the ATE bounds. For institutional

delivery, the largest deviations of the point-estimated ATEs from the lower limit of its ATE bound is

observed.17 We find consistent results even when we used different covariate sets that have been used in

two other previous studies.

Overall, this study provided sufficiently strong evidence that the point-estimated ATEs could have been

biased in previous studies. This study re-estimated the causal impacts through a partial identification

approach and shows the conservative bounds of ATEs. Also, this study quantified how much at least

the point-identified ATEs under questionable identification assumptions could be far from the true ATE.

Certainly, the ATE bounds themselves do not give definite values of the causal impacts. We believe,

however, that the honest estimation of the bounds based on the credible assumptions could be more

useful and valuable in policy making than definitive ATE estimates that heavily rely on unpalatable or

untenable assumptions. We hope the results of this study contribute to evaluating the JSY thoroughly.

17However, it does not necessarily mean that the actual size of under-estimation for the institutional delivery is the largest.It just means that its lower limit of potential underestimation is the largest.

41

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Appendix

A Appendix – Additional results

A.1 Additional evidence in LPSs from the National Family Health Survey (NFHS)

We complement the analysis with the latest National Family Household Survey (NFHS-4) conductedin 2015-2016. In contrast to the DLHS-4, the NFHS is nationally representative, which allows us toestimate the impacts in LPSs. The NFHS is a nationwide household survey that provides informationon health, health-related behaviours and household socio-economic status; it is the Indian version of theDemographic Health Survey conducted in more than 85 low- and middle-income countries (Corsi et al.,2012). The NFHS-4 has 19,906 female observations in LPSs.18 After dropping the observations withmissing information19, our final sample size has become 13,112. Descriptive statistics for the NFHS-4 areshown in Table A.1.

Table A.1: Descriptive statistics in LPSs of the NFHS-4Non-participants

LPSsParticipants

LPSsmean mean

Institutional delivery 0.80 1.00Skilled birth attendance 0.82 0.99Antenatal care at least once 0.86 0.95Antenatal care (≥3 times) 0.76 0.87Postnatal care for mother 0.72 0.87Postnatal care for baby 0.35 0.48Iron and folic acid (IFA) supplement 0.83 0.92Tetanus toxoid (TT) injection 0.89 0.94Below the poverty line card 0.31 0.38Scheduled caste 0.16 0.24Scheduled tribe 0.25 0.30Rural 0.65 0.69Birth order (parity) 2.18 1.90Hindu 0.64 0.65Maternal age 27.62 26.88Maternal education years 3.55 3.72Paternal education years 3.73 3.74Wealth 3.38 2.03

Observations 10783 2338Source: NFHS-4.

A.1.1 Point-identification in LPSs

In this subsection, we show the results for the LPSs obtained from the NFHS-4 data. In contrast to theDLHS-4, the NFHS-4 collected data in LPSs as well as HPSs (Figure 1). To make the results comparableto the ones obtained from the DLHS-4 data as much as possible, we use the same covariate sets used for

18In the NFHS data, not all women in the samples were interviewed about their husband’s characteristics. In this study,we use the nationally representative female samples in which information about a woman’s husband is also collected.

19We dropped 67 observations with missing information about the JSY status and 68 observations that do not haveinformation about paternal educational background.

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the DLHS-4, i.e., the covariate set used in Rahman and Pallikadavath (2018).20 The only difference interms of covariates is the component of household wealth. In both data sets, wealth is created from theprincipal component analysis from indicators of asset ownership, housing characteristics and water andsanitation facilities, but their components are different. In the NLHS-4, we use the wealth which wasofficially derived in collaboration with the World Bank, and it has been shown to be a consistent proxy forhousehold income and expenditure (Rutstein and Staveteig, 2014; Montgomery et al., 2000).21 Althoughin LPSs, all women are eligible for the JSY regardless of their age, socio-economic status, or number ofchildren, we continue to use the same eligibility criteria that are applied in HPSs as an instrument, whichmakes the results comparable with the previous results in HPSs.

Table A.2 reports the ATEs estimated by: (1) Independence, (2) OLS, (3) PSM, (4) IPW+OLS, (5)NNM, and (6) EBW. All the point-estimated ATEs are positive and significant (p < 0.01). We find muchlarger ATEs in LPSs on institutional delivery and skilled birth attendance than ATEs in HPSs estimatedfrom the DLHS-4 data, which could be attributed to the observation that in LPSs the initial outcomelevels are far lower than those in HPSs. The lower initial level and wider coverage of the programme mayhave led to larger ATEs (Dongre, 2012).

A.1.2 Partial identification in LPSs

First, we surmise the direction of the selection bias. Comparing the ATE under the independence as-sumption and the ATE estimated by OLS, we find that the ATEs under the independence assumptionare larger than those estimated by OLS for ANC at least once, PNC for babies, and iron and folic acidsupplement intakes and tetanus toxoid injections. Hence for these outcomes, we impose the positive MTSassumption. For the other outcomes, namely institutional delivery, skilled birth attendance, ANC threetimes and more, and PNC for mothers, the negative MTS assumption is imposed.

Table A.3 shows the ATE bounds in LPSs estimated from the NFHS-4 data. Figure A.1 visually sum-marises the bound-estimated ATEs and the point-estimated ATEs. Overall, we find that the point-identified ATEs are outside of the ATE bounds for institutional delivery, skilled birth attendance, ANCat least once, ANC three times and more and PNC for mothers. Compared with the results in HPSs,the deviations of the point-identified ATEs from the upper/lower limits of ATE bounds are substantiallysmaller in LPSs, suggesting the lower limits of the selection bias in LPSs. For PNC for babies, iron andfolic acid supplement intakes and tetanus toxoid injections, some of the ATEs estimated under the CIAare included within the ATE bounds. In particular, for the iron and folic acid supplement intakes, all thepoint-identified ATEs are within the bound, which is attributable to the weak identification power of theMIV assumption. Additional imposition of the MIV little contributes to tightening the ATE bound.

Overall, we find that most of the ATEs under the CIA are outside the ATE bound. The exception isthe effect on the iron and folic acid supplement intakes in which all the point-identified ATEs are insidethe partially-identified ATE bound. Comparing the results in both HPSs and LPSs, we observe that thedeviations from the ATE bounds are larger in HPSs. The contrast between HPSs and LPSs suggests thatwe are more likely to suffer larger selection bias when the eligibility of the JSY is restricted to marginalisedmothers.

20We also estimate the ATEs with the covariate sets in Lim et al. (2010) and Sengupta and Sinha (2018), and we obtainedvery similar results.

21The list of indicators can be downloaded from the official website (https://www.dhsprogram.com/programming/wealth%20index/India%20DHS%202015-16/IndiaNFHS4.pdf)–Accessed in 14/02/2019–.

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Table A.2: Point-estimated ATE in LPSs from the NFHS-4Estimates SEs Confidence intervals

Institutional deliveryIndependence 0.432*** 0.005 (0.422, 0.442)OLS 0.433*** 0.004 (0.425, 0.441)PSM 0.435*** 0.005 (0.425, 0.445)IPW+OLS 0.433*** 0.004 (0.425, 0.441)NNM 0.428*** 0.005 (0.418, 0.438)EBW 0.429*** 0.005 (0.419, 0.439)

Skilled delivery attendanceIndependence 0.349*** 0.005 (0.339, 0.359)OLS 0.35*** 0.005 (0.34, 0.36)PSM 0.35*** 0.005 (0.34, 0.36)IPW+OLS 0.35*** 0.005 (0.34, 0.36)NNM 0.346*** 0.005 (0.336, 0.356)EBW 0.345*** 0.005 (0.335, 0.355)

Antenatal care +1Independence 0.105*** 0.005 (0.095, 0.115)OLS 0.105*** 0.005 (0.095, 0.115)PSM 0.104*** 0.006 (0.092, 0.116)IPW+OLS 0.105*** 0.005 (0.095, 0.115)NNM 0.097*** 0.006 (0.085, 0.109)EBW 0.095*** 0.005 (0.085, 0.105)

Antenatal care +3Independence 0.101*** 0.007 (0.087, 0.115)OLS 0.104*** 0.007 (0.09, 0.118)PSM 0.105*** 0.008 (0.089, 0.121)IPW+OLS 0.104*** 0.007 (0.09, 0.118)NNM 0.093*** 0.008 (0.077, 0.109)EBW 0.099*** 0.007 (0.085, 0.113)

Postnatal care for motherIndependence 0.178*** 0.007 (0.164, 0.192)OLS 0.18*** 0.007 (0.166, 0.194)PSM 0.176*** 0.008 (0.16, 0.192)IPW+OLS 0.18*** 0.007 (0.166, 0.194)NNM 0.178*** 0.008 (0.162, 0.194)EBW 0.178*** 0.007 (0.164, 0.192)

Postnatal care for babyIndependence 0.065*** 0.007 (0.051, 0.079)OLS 0.061*** 0.007 (0.047, 0.075)PSM 0.055*** 0.008 (0.039, 0.071)IPW+OLS 0.06*** 0.007 (0.046, 0.074)NNM 0.058*** 0.008 (0.042, 0.074)EBW 0.064*** 0.007 (0.05, 0.078)

Iron folic acid supplement intakesIndependence 0.104*** 0.006 (0.092, 0.116)OLS 0.095*** 0.006 (0.083, 0.107)PSM 0.093*** 0.007 (0.079, 0.107)IPW+OLS 0.095*** 0.006 (0.083, 0.107)NNM 0.086*** 0.007 (0.072, 0.1)EBW 0.091*** 0.006 (0.079, 0.103)

Tetanus toxoid injectionsIndependence 0.056*** 0.004 (0.048, 0.064)OLS 0.052*** 0.004 (0.044, 0.06)PSM 0.052*** 0.004 (0.044, 0.06)IPW+OLS 0.052*** 0.004 (0.044, 0.06)NNM 0.05*** 0.004 (0.042, 0.058)EBW 0.047*** 0.003 (0.041, 0.053)

Source: NFHS-4. Note: Number of observations is 13,112. The choice of covariates follows Rahman and Pallikadavath(2018). PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverse probabilityweighting; NNM=Nearest neighbour matching; and EBW=Entropy balance weighting. * p < 0.10, ** p < 0.05, ***p < 0.01.

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Table A.3: ATE bounds in LPSs estimated from the NFHS-4Estimates Confidence intervals

Lower bound Upper bound Lower bound Upper boundInstitutional delivery

No assumption −0.319 0.681 −0.325 0.686MTR 0.000 0.681 0.000 0.686MTR+MTS 0.432 0.681 0.425 0.686MTR+MTS+MIV 0.505 0.624 0.494 0.632

Skilled delivery attendanceNo assumption −0.364 0.636 −0.369 0.642MTR 0.000 0.636 0.000 0.642MTR+MTS 0.349 0.636 0.341 0.642MTR+MTS+MIV 0.413 0.583 0.402 0.592

Antenatal care +1No assumption −0.487 0.513 −0.493 0.519MTR 0.000 0.513 0.000 0.519MTR+MTS 0.000 0.105 0.000 0.114MTR+MTS+MIV 0.000 0.086 0.000 0.098

Antenatal care +3No assumption −0.460 0.540 −0.466 0.547MTR 0.000 0.540 0.000 0.547MTR+MTS 0.101 0.540 0.089 0.547MTR+MTS+MIV 0.130 0.500 0.114 0.510

Postnatal care for motherNo assumption −0.432 0.568 −0.437 0.574MTR 0.000 0.568 0.000 0.574MTR+MTS 0.178 0.568 0.167 0.574MTR+MTS+MIV 0.219 0.521 0.205 0.531

Postnatal care for babyNo assumption −0.449 0.551 −0.455 0.557MTR 0.000 0.551 0.000 0.557MTR+MTS 0.000 0.065 0.000 0.075MTR+MTS+MIV 0.000 0.056 0.000 0.069

Iron folic acid supplement intakesNo assumption −0.477 0.523 −0.483 0.529MTR 0.000 0.523 0.000 0.529MTR+MTS 0.000 0.104 0.000 0.114MTR+MTS+MIV 0.000 0.103 0.000 0.114

Tetanus toxoid injectionsNo assumption −0.526 0.474 −0.532 0.480MTR 0.000 0.474 0.000 0.480MTR+MTS 0.000 0.056 0.000 0.061MTR+MTS+MIV 0.000 0.050 0.000 0.058

Source: NFHS-4. Note: Number of observations is 13,112. 95% confidence intervals are calculated following Imbens andManski (2004) by bootstrap with 200 repetitions.

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Figure A.1: ATE bound and point-estimated ATE in LPSs from the NFHS-4

Source: NFHS-4. Note: The choice of covariates follows Rahman and Pallikadavath (2018).PSM=Propensity score matching; IPW+OLS=OLS adjusted with the propensity score inverse probability weighting;NNM=Nearest neighbour matching; EBW=Entropy balance weighting; ANC=Antenatal care; PNC=Postnatal care;IFA=Iron and folic acid; and TT=tetanus toxoid. The bounds shown are sharp bounds estimated under the MTR, MTSand MIV assumptions. The shaded areas show the 95% level confidence interval of the ATE bound under the MTR, MTSand MIV assumptions.

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B Appendix – Note on methodology

B.1 Entropy balance weighting

In the non-experimental study, wherein participation in treatment is not randomised, the pure compari-son between treated people and untreated people does not generally provide the causal treatment effectsbecause those who are treated can be fundamentally different from the untreated people with regards totheir characteristics, such as the educational background. An entropy balance weighting, developed byHainmueller (2012), perfectly balances the moments of the covariate distributions.

The entropy balance weighting method assigns weights such that the covariate distributions in theweighted data satisfy a set of moment conditions set by a researcher. Specifically, weights are assignedto the control group so that the moments of the weighted covariate distributions in the control groupmatch those in the treatment group. The entropy balance weighting method has a few advantages overthe propensity score weighting. Firstly, the entropy balance weighting can “perfectly” balance a set ofmoments of the covariate distributions between two groups, while the propensity score methods often failto jointly balance all the covariate distributions in practice partly because of the misspecification of thepropensity score. In order to improve the balance of some covariates, the propensity score method usuallyneeds to sacrifice the balance of the other covariates (Iacus et al., 2012). Thanks to the perfect balancingobtained with the entropy balancing method, manual balance checking is no longer necessary. Hence,we don’t have to go back and forth between estimating the propensity score and manually checking thebalance of the covariates (Hainmueller, 2012).

Each observation in the untreated group gets a weight satisfying a set of balance constraints set bythe researcher a priori. Entropy balancing estimates the weights directly from a set of balance constraintsand the normalisation and non-negativity constraints. The optimal weights for the untreated group arechosen in such a way that they minimise the following entropy distance metric:

H(w) =∑

{i|Untreated}

wiln(wi/qi), (B.1)

subject to the following balance and normalisation constraints:∑{i|Untreated}

wicri(Xi) = mr with r ∈ {1, ..., R}, (B.2)

∑{i|Untreated}

wi = 1, and (B.3)

wi ≥ 0, ∀i ∈ (Untreated). (B.4)

Equation (B.2) is a set of balance constraints. Equations (B.3) and (B.4) are the normalisation constraintand the non-negativity constraint respectively. The part wicri(Xi) = mr in equation (B.2) describes aset of R balance constraints imposed on the moments of the weighted distributions of the covariates inthe control group. Each balance constraint equates a certain order moment of the weighted covariatedistributions in the control group to the corresponding moment of the covariate distributions in thetreated group. As the entropy balancing can balance the higher order moments of the distributions, themoment constraints can be not only the mean (first moment), but also the variance (second moment),the skewness (third moment), and beyond them. For example, if we are interested in estimating theweights for balancing the rth order moment of a specific covariate, say Xp, we set cri(Xip) = (Xip)

r orcri(Xip) = (Xip − µp)r with mean µp. The choice of the moment order can vary across the covariatesand largely depends on the researcher’s a priori knowledge about their distribution types. For example,

53

if the covariates are binary variables, the first order moment balancing is sufficient, whereas in the caseof the variables that are normally distributed, balancing the first and second order moments is sufficientto balance their entire distributions. In this study, we will balance the covariate distributions up totheir second moments. Using the estimated weight, we non-parametrically estimate the means and thequantiles of the distributions of Y0 and Y1.

B.2 MTS sharp bound and independence

This note shows why the upper bound of ATE under the positive MTS assumption corresponds to theATE under the independence assumption. This note is based on the explanation made by McCarthyet al. (2015). Since the upper bound of ATE is, by definition, the difference between the upper bound ofP (Y1 = 1) and the lower bound of P (Y0 = 1), we firstly discuss in what circumstance the upper bound ofP (Y1 = 1) and the lower bound of P (Y0 = 1) are attained.

Applying the law of total probability for P (Y1 = 1),

P (Y1 = 1) = P (Y1 = 1|D = 1)P (D = 1) + P (Y1 = 1|D = 0)P (D = 0)

= P (Y1 = 1|D = 1){1− P (D = 0)}+ P (Y1 = 1|D = 0)P (D = 0)

= {P (Y1 = 1|D = 0)− P (Y1 = 1|D = 1)}P (D = 0) + P (Y1 = 1|D = 1). (B.5)

Under the MTS assumption, the term in the bracket is non-positive. The upper bound of P (Y1 = 1),which is P (Y1 = 1|D = 1), is achieved when the term in the bracket becomes 0, where the independenceassumption, P (Y1 = 1|D = 0) = P (Y1 = 1|D = 1), is satisfied.

Next, applying the law of total probability for P (Y0 = 1),

P (Y0 = 1) = P (Y0 = 1|D = 1)P (D = 1) + P (Y0 = 1|D = 0)P (D = 0)

= P (Y0 = 1|D = 1)P (D = 1) + P (Y0 = 1|D = 0){1− P (D = 1)}= {P (Y0 = 1|D = 1)− P (Y0 = 1|D = 0)}P (D = 1) + P (Y0 = 1|D = 0), (B.6)

Under the MIS assumption, the term in the bracket is non-negative. The lower bound of P (Y0 = 1),which is P (Y0 = 1|D = 0), is achieved when the term in the bracket becomes 0, where the independenceassumption, P (Y0 = 1|D = 1) = P (Y0 = 1|D = 0), is satisfied.

Combined them together, the upper limit of ATE, which is P (Y1 = 1|D = 1) − P (Y0 = 1|D = 0), isachieved when P (Yt = 1|D = 1) = P (Yt = 1|D = 0) = P (Yt = 1) holds for t ∈ {0, 1}. This condition isindeed the independence assumption. In the same way, we can show that the lower bound of ATE underthe negative MTS assumption corresponds to the ATE estimated under the independence assumption.

B.3 Derivation of the MIV bounds in equations (29) and (30)

First, for t ∈ {0, 1} and t′ 6= t, the bound of the conditional probability, P (Yt = 1|v = u) is given by

P (Yt = 1|D = t, v = u)P (D = t|v = u)

≤ P (Yt = 1|v = u) (B.7)

≤ {P (Yt = 1|D = t, v = u)P (D = t|v = u) + P (D = t′|v = u)}.

54

Under the MIV assumption, for u1 ≤ u ≤ u2,

maxu1≤u

P (Yt = 1|D = t, v = u1)P (D = t|v = u1)

≤ P (Yt = 1|v = u) (B.8)

≤ minu≤u2{P (Yt = 1|D = t, v = u2)P (D = t|v = u2) + P (D = t′|v = u2)}.

Applying the law of total probability, we obtain the bounds of P (Yt = 1) as follows:∑u∈V

P (v = u) maxu1≤u

P (Yt = 1|D = t, v = u1)P (D = t|v = u1)

≤ P (Yt = 1) (B.9)

≤∑u∈V

P (v = u) minu≤u2{P (Yt = 1|D = t, v = u2)P (D = t|v = u2) + P (D = t′|v = u2)}.

When an instrument is binary, we can simplify this. As u, u0 and u1 take the values of 0 or 1,

P (v = 0)P (Yt = 1|D = t, v = 0)P (D = t|v = 0)

+P (v = 1) max{P (Yt = 1|D = t, v = 0)P (D = t|v = 0), P (Yt = 1|D = t, v = 1)P (D = t|v = 1)}≤ P (Yt = 1) (B.10)

≤ P (v = 0) min{P (Yt = 1|D = t, v = 0)P (D = t|v = 0) + P (D = t′|v = 0),

P (Yt = 1|D = t, v = 1)P (D = t|v = 1) + P (D = t′|v = 1)}+

P (v = 1)[P (Yt = 1|D = t, v = 1)P (D = t|v = 1) + P (D = t′|v = 1)].

This leads to equations (29) and (30).

B.4 Finite-sample bias in the presence of maxima and minima operators

First, for t ∈ {0, 1} and t′ 6= t, we show the estimate of the lower bound of P (Yt = 1|v = u) has an upwardbias.

When the estimate of P (.) is denoted by P (.), the estimated lower bound is expressed as

maxu1≤u

P (Yt = 1|D = t, v = u1)P (D = t|v = u1). (B.11)

Since a maximum operator is a convex function, by the Jensen’s inequality,

E[maxu1≤u

P (Yt = 1|D = t, v = u1)P (D = t|v = u1)]

≥ maxu1≤u

E[P (Yt = 1|D = t, v = u1)P (D = t|v = u1)]

= maxu1≤u

P (Yt = 1|D = t, v = u1)P (D = t|v = u1). (B.12)

Equality in the last holds due to the unbiasedness of P (.). Hence equation (B.12) shows that the estimateof the lower bound can have a upward bias.

Next, we show the estimate of the upper bound of P (Yt = 1|v = u) has an upward bias. The esti-mated upper bound is expressed as

minu≤u0{P (Yt = 1|D = t, v = u0)P (D = t|v = u0) + P (D = t′|v = u0)}. (B.13)

55

Since a maximum operator is a concave function, by the Jensen’s inequality,

E[minu≤u0{P (Yt = 1|D = t, v = u0)P (D = t|v = u0) + P (D = t′|v = u0)}]

≤ minu≤u0

E[{P (Yt = 1|D = t, v = u0)P (D = t|v = u0) + P (D = t′|v = u0)}]

= minu≤u0{P (Yt = 1|D = t, v = u0)P (D = t|v = u0) + P (D = t′|v = u0)}. (B.14)

Equation (B.14) shows that the estimate of the upper bound can have a downward bias.

C Appendix – Derivations of ATE bounds under joint assumptions

C.1 MTR+MTS

We consider the case in which the MTR and MTS assumptions are jointly imposed. Imposing the MTRassumption and the MTS assumption jointly means that we are assuming the intersection of the boundsof P (Y1 = 1) and P (Y0 = 1) that are derived above under each assumption.

(i) MTR+ Positive MTS: If we combine the MTR assumption and the positive MTS assumption,the bounds of P (Y1 = 1) and P (Y0 = 1) become

P (Y = 1)︸ ︷︷ ︸MTR

≤ P (Y1 = 1) ≤ P (Y1 = 1|D = 1)︸ ︷︷ ︸Positive MTS

(C.15)

P (Y0 = 1|D = 0)︸ ︷︷ ︸Positive MTS

≤ P (Y0 = 1) ≤ P (Y = 1)︸ ︷︷ ︸MTR

, (C.16)

where for t ∈ {0, 1} the bound of P (Yt = 1) is an intersection of the corresponding bounds under theMTR assumption and the positive MTS assumption. The new bound of ATE is given by

0 ≤ ATE ≤ P (Y1 = 1|D = 1)− P (Y0 = 1|D = 0). (C.17)

The upper bound of ATE is given by the upper bound under the positive MTS assumption, while thelower bound is 0 thanks to the MTR assumption.

(ii) MTR+ Negative MTS: The combination of the MTR assumption and the negative MTS as-sumption can be obtained in the same fashion.

max{P (Y = 1)︸ ︷︷ ︸MTR

, P (Y1 = 1|D = 1)︸ ︷︷ ︸Negative MTS

} ≤ P (Y1 = 1) ≤ P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)︸ ︷︷ ︸Worst case

(C.18)

P (Y0 = 1|D = 0)P (D = 0)︸ ︷︷ ︸Worst case

≤ P (Y0 = 1) ≤ min{P (Y = 1)︸ ︷︷ ︸MTR

, P (Y0 = 1|D = 0)︸ ︷︷ ︸Negative MTS

} (C.19)

and the ATE is given by

max{P (Y = 1), P (Y1 = 1|D = 1)} −min{P (Y = 1), P (Y0 = 1|D = 0)}≤ ATE (C.20)

≤ {P (Y1 = 1|D = 1)P (D = 1) + P (D = 0)} − P (Y0 = 1|D = 0)P (D = 0),

where the upper bound corresponds to that in the worst-case.

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C.2 MTR+MTS+MIV

We estimate the bounds under the MTR, MTS and MIV assumptions. When we impose the MIV assump-tion, the sharp bounds of P (Y1 = 1) and P (Y0 = 1) cannot be expressed simply by the intersection of thebounds estimated under each of assumption. We firstly derive the bounds for conditional probabilitiesP (Y1 = 1|v = u) and P (Y0 = 1|v = u).

(i) MTR+ positive MTS+ MIV: Under the MTR and positive MTS assumptions, the bounds ofthe conditional probabilities P (Y1 = 1|v = u) and P (Y0 = 1|v = u) are give by

P (Y = 1|v = u)︸ ︷︷ ︸MTR

≤ P (Y1 = 1|v = u) ≤ P (Y1 = 1|D = 1, v = u)︸ ︷︷ ︸Positive MTS

(C.21)

P (Y0 = 1|D = 0, v = u)︸ ︷︷ ︸Positive MTS

≤ P (Y0 = 1|v = u) ≤ P (Y = 1|v = u)︸ ︷︷ ︸MTR

. (C.22)

Under the MIV assumption, for u1 ≤ u ≤ u2,

maxu1≤u

P (Y = 1|v = u1) ≤ P (Y1 = 1|v = u) ≤ minu≤u2

P (Y1 = 1|D = 1, v = u2) (C.23)

maxu1≤u

P (Y0 = 1|D = 0, v = u1) ≤ P (Y0 = 1|v = u) ≤ minu≤u2

P (Y = 1|v = u2). (C.24)

Applying the law of total probability, we obtain the bounds of P (Y1 = 1) and P (Y0 = 1).∑u∈V

P (v = u) maxu1≤u

P (Y = 1|v = u1)

≤ P (Y1 = 1) (C.25)

≤∑u∈V

P (v = u) minu≤u2

P (Y1 = 1|D = 1, v = u2)

and ∑u∈V

P (v = u) maxu1≤u

P (Y0 = 1|D = 0, v = u1)

≤ P (Y0 = 1) (C.26)

≤∑u∈V

P (v = u) minu≤u2

P (Y = 1|v = u2).

When an instrument is binary, we can simplify them. As u, u0 and u1 take the values of 0 or 1, arrangingthese equations lead to equations (C.27) and (C.28).

P (v = 0)P (Y = 1|v = 0) + P (v = 1) max{P (Y = 1|v = 0), P (Y = 1|v = 1)}≤ P (Y1 = 1) (C.27)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0), P (Y1 = 1|D = 1, v = 1)}+ P (v = 1)P (Y1 = 1|D = 1, v = 1)

and

P (v = 0)P (Y0 = 1|D = 0, v = 0) + P (v = 1) max{P (Y0 = 1|D = 0, v = 0), P (Y0 = 1|D = 0, v = 1)}≤ P (Y0 = 1) (C.28)

≤ P (v = 0) min{P (Y = 1|v = 0), P (Y = 1|v = 1)}+ P (v = 1)P (Y = 1|v = 1).

The sharp bound of the ATE can be obtained by equation (8).

57

(i) MTR+ negative MTS+ MIV: Under the MTR, negative MTS and MIV assumptions, the boundsof the conditional probabilities P (Y1 = 1|v = u) and P (Y0 = 1|v = u) are

max{P (Y = 1|v = u)︸ ︷︷ ︸MTR

, P (Y1 = 1|D = 1, v = u)︸ ︷︷ ︸Negative MTS

}

≤ P (Y1 = 1|v = u) (C.29)

≤ P (Y1 = 1|D = 1, v = u)P (D = 1|v = u) + P (D = 0|v = u)

and

maxu1≤u

P (Y0 = 1|D = 0, v = u)P (D = 0|v = u)

≤ P (Y0 = 1|v = u) (C.30)

≤ minu≤u2{P (Y = 1|v = u)︸ ︷︷ ︸

MTR

, P (Y0 = 1|D = 0, v = u)︸ ︷︷ ︸Negative MTS

}.

Under the MIV assumption, for u1 ≤ u ≤ u2,

maxu1≤u

max{P (Y = 1|v = u1), P (Y1 = 1|D = 1, v = u1)}

≤ P (Y1 = 1|v = u) (C.31)

≤ minu≤u2

[P (Y1 = 1|D = 1, v = u2)P (D = 1|v = u2) + P (D = 0|v = u2)]

and

maxu1≤u

P (Y0 = 1|D = 0, v = u1)P (D = 0|v = u1)

≤ P (Y0 = 1|v = u) (C.32)

≤ minu≤u2

min{P (Y = 1|v = u2), P (Y0 = 1|D = 0, v = u2)}.

Applying the law of total probability,∑u∈V

P (v = u) maxu1≤u

max{P (Y = 1|v = u1), P (Y1 = 1|D = 1, v = u1)}

≤ P (Y1 = 1) (C.33)

≤∑u∈V

P (v = u) minu≤u2{P (Y1 = 1|D = 1, v = u2)P (D = 1|v = u2) + P (D = 0|v = u2)}

and ∑u∈V

P (v = u) maxu1≤u

P (Y0 = 1|D = 0, v = u1)P (D = 0|v = u1)

≤ P (Y0 = 1) (C.34)

≤∑u∈V

P (v = u) minu≤u2

min{P (Y = 1|v = u2), P (Y0 = 1|D = 0, v = u2)}.

As u, u1 and u2 take the values of 0 or 1, arranging them leads to equations (C.35) and (C.36).

58

P (v = 0) max{P (Y = 1|v = 0), P (Y1 = 1|D = 1, v = 0)}+

P (v = 1) max{P (Y = 1|v = 0), P (Y1 = 1|D = 1, v = 0), P (Y = 1|v = 1), P (Y1 = 1|D = 1, v = 1)}≤ P (Y1 = 1) (C.35)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0) + P (D = 0|v = 0),

P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}+P (v = 1){P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}

and

P (v = 0)P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0)

+P (v = 1) max{P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0), P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1)}≤ P (Y0 = 1) (C.36)

≤ P (v = 0) min{P (Y = 1|v = 0), P (Y0 = 1|D = 0, v = 0), P (Y = 1|v = 1), P (Y0 = 1|D = 0, v = 1)}+P (v = 1) min{P (Y = 1|v = 1), P (Y0 = 1|D = 0, v = 1)}.

The sharp bound of the ATE can be obtained by equation (8).

D Appendix – Derivations of ATE under the other combinations ofassumptions

D.1 MTS+MIV

(i) Positive MTS+MIV: Under the positive MTS assumption, the bounds of the conditional proba-bilities P (Y1 = 1|v = u) and P (Y0 = 1|v = u) are give by

P (Y1 = 1|D = 1, v = u)P (D = 1|v = u)

≤ P (Y1 = 1|v = u) (D.37)

≤ P (Y1 = 1|D = 1, v = u)

and

P (Y0 = 1|D = 0, v = u)

≤ P (Y0 = 1|v = u) (D.38)

≤ P (Y0 = 1|D = 0, v = u) + P (D = 1|v = u).

Applying the law of total probability, we obtain the bounds of P (Y1 = 1) and P (Y0 = 1) as follows:∑u∈V

P (v = u) maxu1≤u

P (Y1 = 1|D = 1, v = u1)P (D = 1|v = u1)

≤ P (Y1 = 1) (D.39)

≤∑u∈V

P (v = u) minu≤u2

P (Y1 = 1|D = 1, v = u2)

59

and ∑u∈V

P (v = u) maxu1≤u

P (Y0 = 1|D = 0, v = u1)

≤ P (Y0 = 1) (D.40)

≤∑u∈V

P (v = u) minu≤u2{P (Y0 = 1|D = 0, v = u2) + P (D = 1|v = u2)}.

When an instrument is binary, we can simplify them. As u, u0 and u1 take the values of 0 or 1,

P (v = 0)P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0) +

P (v = 1) max{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0), P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1)}≤ P (Y1 = 1) (D.41)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0), P (Y1 = 1|D = 1, v = 1)}+

P (v = 1)P (Y1 = 1|D = 1, v = 1)

and

P (v = 0)P (Y0 = 1|D = 0, v = 0) +

P (v = 1) max{P (Y0 = 1|D = 0, v = 0), P (Y0 = 1|D = 0, v = 1)}≤ P (Y0 = 1) (D.42)

≤ P (v = 0) min{P (Y0 = 1|D = 0, v = 0) + P (D = 1|v = 0), P (Y0 = 1|D = 0, v = 1) + P (D = 1|v = 1)}+

P (v = 1){P (Y0 = 1|D = 0, v = 1) + P (D = 1|v = 1)}.

(ii) Negative MTS+MIV: Under the negative MTS assumption, the bounds of the conditional prob-abilities P (Y1 = 1|v = u) and P (Y0 = 1|v = u) are give by

P (Y1 = 1|D = 1, v = u)

≤ P (Y1 = 1|v = u)

≤ P (Y1 = 1|D = 1, v = u)P (D = 1|v = u) + P (D = 0|v = u) (D.43)

and

P (Y0 = 1|D = 0, v = u)P (D = 0|v = u)

≤ P (Y0 = 1|v = u)

≤ P (Y0 = 1|D = 0, v = u). (D.44)

Applying the law of total probability, we obtain the bounds of P (Y1 = 1) and P (Y0 = 1) as follows:∑u∈V

P (v = u) maxu1≤u

P (Y1 = 1|D = 1, v = u1)

≤ P (Y1 = 1) (D.45)

≤∑u∈V

P (v = u) minu≤u2{P (Y1 = 1|D = 1, v = u2)P (D = 1|v = u2) + P (D = 0|v = u2)}

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and ∑u∈V

P (v = u) maxu1≤u

P (Y0 = 1|D = 0, v = u1)P (D = 0|v = u1)

≤ P (Y0 = 1) (D.46)

≤∑u∈V

P (v = u) minu≤u2

P (Y0 = 1|D = 0, v = u2).

When an instrument is binary, we can simplify them. As u, u0 and u1 take the values of 0 or 1,

P (v = 0)P (Y1 = 1|D = 1, v = 0) +

P (v = 1) max{P (Y1 = 1|D = 1, v = 0), P (Y1 = 1|D = 1, v = 1)}≤ P (Y1 = 1) (D.47)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0) + P (D = 0|v = 0),

P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}+

P (v = 1){P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}

and

P (v = 0)P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0) +

P (v = 1) max{P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0), P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1)}≤ P (Y0 = 1) (D.48)

≤ P (v = 0) min{P (Y0 = 1|D = 0, v = 0), P (Y0 = 1|D = 0, v = 1)}+

P (v = 1)P (Y0 = 1|D = 0, v = 1).

D.2 MTR+MIV

Under the MTR assumption, the bounds of the conditional probabilities P (Y1 = 1|v = u) and P (Y0 =1|v = u) are give by

P (Y = 1|v = u)

≤ P (Y1 = 1|v = u) (D.49)

≤ P (Y1 = 1|D = 1, v = u)P (D = 1|v = u) + P (D = 0|v = u)

and

P (Y0 = 1|D = 0, v = u)P (D = 0|v = u)

≤ P (Y0 = 1|v = u) (D.50)

≤ P (Y = 1|v = u).

Applying the law of total probability, we obtain the bounds of P (Y1 = 1) and P (Y0 = 1) as follows:∑u∈V

P (v = u) maxu1≤u

P (Y = 1|v = u1)

≤ P (Y1 = 1) (D.51)

≤∑u∈V

P (v = u) minu≤u2

[P (Y1 = 1|D = 1, v = u2)P (D = 1|v = u2) + P (D = 0|v = u2)]

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and ∑u∈V

P (v = u) maxu1≤u

P (Y0 = 1|D = 0, v = u1)P (D = 0|v = u1)

≤ P (Y0 = 1) (D.52)

≤∑u∈V

P (v = u) minu≤u2

P (Y = 1|v = u2).

When an instrument is binary, we can simplify them. As u, u0 and u1 take the values of 0 or 1,

P (v = 0)P (Y = 1|v = 0) + P (v = 1) max{P (Y = 1|v = 0), P (Y = 1|v = 1)}≤ P (Y1 = 1) (D.53)

≤ P (v = 0) min{P (Y1 = 1|D = 1, v = 0)P (D = 1|v = 0) + P (D = 0|v = 0),

P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}+P (v = 1){P (Y1 = 1|D = 1, v = 1)P (D = 1|v = 1) + P (D = 0|v = 1)}

and

P (v = 0)P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0) +

P (v = 1) max{P (Y0 = 1|D = 0, v = 0)P (D = 0|v = 0), P (Y0 = 1|D = 0, v = 1)P (D = 0|v = 1)}≤ P (Y0 = 1) (D.54)

≤ P (v = 0) min{P (Y = 1|v = 0), P (Y = 1|v = 1)}+ P (v = 1)P (Y = 1|v = 1).

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